Levels of p-adic automorphic forms and a p-adicJacquet-Langlands correspondence
A thesis presented for the degree of
Doctor of Philosophy of Imperial College London
and the
Diploma of Imperial College by
James Newton
Department of Mathematics
Imperial College London
180 Queen’s Gate, London SW7 2BZ
JUNE 30, 2011
2
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fully acknowledged in accordance with the standard referencing practices of the discipline.
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Abstract
We investigate the arithmetic of p-adic automorphic forms for certain quaternion algebras
over totally real fields (including GL2/Q), focussing on the question of the relationships
between p-adic automorphic forms with different levels at a prime l different from p, and
the relation between p-adic automorphic forms for (the multiplicative groups of) different
quaternion algebras (i.e. a p-adic Jacquet–Langlands correspondence). In chapters 2 and 3
we prove results of ‘level raising’ type, showing that certain families of p-adic automorphic
forms with level prime to l (l-old forms) intersect with a family of p-adic automorphic
form with Iwahori level at l, where all the classical points in this second family are l-new.
Chapter 2 works with definite quaternion algebras over Q, whilst chapter 3 works with
GL2/Q. In chapter 3 the main tool is Emerton’s theory of completed cohomology. In
chapter 4 we study indefinite quaternion algebras over totally real fields F , split at one
infinite place, and prove level raising and lowering results. Finally, also in chapter 4, we
give an example of a cohomological construction of p-adic Jacquet-Langlands functoriality,
using completed cohomology.
5
Acknowledgements
This thesis is the result of research carried out under the supervision of Kevin Buzzard,
to whom I am grateful for providing such excellent guidance. Chapters 3 and 4 crucially
rely on work of Matthew Emerton, whom I also thank for several helpful conversations and
communicating details of an earlier draft of [20]. Chapters 2 and 3 have benefited from
comments from anonymous referees, Owen Jones, David Loeffler, Kevin McGerty and
Alex Paulin. I would like to thank the participants and organisers involved with the various
London Number Theory study groups and seminars, from which I have benefited greatly
during my time as a graduate student. The author is supported by an EPSRC doctoral
training grant, and part of this work was done during the author’s stay at the Institut Henri
Poincare – Centre Emile Borel. The writing up process was completed whilst the author
was a member of the Institute for Advanced Study, supported by NSF grant DMS-0635607.
I am grateful to these organisations for their support.
Finally, I would like to thank all my friends and colleagues in London and elsewhere,
especially Andrew for his love and support over the last few years.
6
Table of contents
Abstract 4
1 Introduction 71.1 Eigenvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Levels of modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Jacquet-Langlands correspondence . . . . . . . . . . . . . . . . . . . . . . 91.4 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Level raising for p-adic automorphic forms 122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma . 152.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Completed cohomology and level raising for p-adic modular forms 403.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Completed cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Non-optimal levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Eigencurves of newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5 Addendum: a remark on the level at p . . . . . . . . . . . . . . . . . . . . 50
4 Completed cohomology of Shimura curves 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Shimura curves and their bad reduction . . . . . . . . . . . . . . . . . . . 574.4 Completed cohomology of Shimura curves . . . . . . . . . . . . . . . . . 634.5 Eigenvarieties and an overconvergent Jacquet–Langlands correspondence . 77
References 87
7
Chapter 1
Introduction
This thesis investigates the arithmetic of p-adic automorphic forms for certain quaternion
algebras over totally real fields (including GL2/Q). It comprises three (more or less inde-
pendent) chapters, with common themes: the question of the relationships between p-adic
automorphic forms with different levels at a prime l 6= p, and the relation between p-adic
automorphic forms for (the multiplicative groups of) different quaternion algebras (i.e. a
p-adic Jacquet-Langlands correspondence). Chapter 2 is based on the publication [32],
chapter 3 is based on the publication [33]. In the rest of this introduction, we will take the
opportunity to summarise some previous work in this area, underpinning our research, and
then summarise the results contained in this thesis.
1.1 Eigenvarieties
An eigenvariety is a rigid analytic space, p-adically interpolating the systems of Hecke
eigenvalues arising from certain automorphic forms. The first construction of an eigenva-
riety (which was named ‘the eigencurve’ since it is one-dimensional) occurs in [13], and it
is constructed using Coleman’s theory of overconvergent modular forms [12]. Buzzard ex-
tended this construction to automorphic forms for definite quaternion algebras over totally
real fields [2, 3] and provided a ‘machine’ which has enabled the construction of eigen-
varieties in certain other situations [9, 29]. A related construction is also carried out in
[45].
A rather different approach to the study of p-adic automorphic forms can be found in
[19]. Emerton’s approach has recently led to a proof of many cases of the Fontaine-Mazur
conjecture [20] and has therefore been the subject of much interest. Emerton’s completed
1.2 Levels of modular forms 8
cohomology and construction of eigenvarieties is the object of study in chapters 3 and 4,
where we study p-adic automorphic forms for GL2/Q and indefinite quaternion algebras
over totally real fields, split at one infinite place.
1.2 Levels of modular forms
Much of this work is inspired by work of Diamond, Ribet and Taylor from the 80s and 90s
[37, 44, 38, 14, 15] together with more recent generalisations of Ribet’s work due to Jarvis
and Rajaei [25, 36]. Chapters 2 and 3 are influenced by work on level raising results for
modular forms. These results describe congruences between modular forms of different
levels, for example if f is a weight k cuspidal eigenform of level Γ1(N), with its Hecke
eigenvalues at l satisfying a certain congruence condition modulo a prime p, then there is
a weight k cuspidal eigenform g of level Γ1(N) ∩ Γ0(l) which is l-new and congruent to f
modulo p (more precisely g is congruent to an l-old form of level Γ1(N)∩Γ0(l) constructed
from f ). In Chapters 2 and 3 we prove results in the same spirit, but for families of p-adic
modular forms. This comes down to showing that a family of l-old modular forms intersects
a family of l-new modular forms at a point corresponding to a non-classical p-adic modular
form which is simultaneously l-old and l-new.
On the other hand one can consider level lowering results for modular forms. These are
of the following type: given a weight k cuspidal eigenform g of level Γ1(N) ∩ Γ0(l), such
that the mod p Galois representation
ρg : Gal(Q/Q)→ GL2(Fp)
attached to g is irreducible and unramified at l, then there should be a weight k cuspidal
eigenform f of level Γ1(N) such that ρf ∼= ρg. Chapter 4 investigates these questions in
the setting of p-adic automorphic forms for indefinite quaternion algebras over totally real
fields, split at one infinite place (so the relevant Shimura varieties are one-dimensional)
— we will be interested in the situation where the characteristic 0 Galois representation
attached to a p-adic automorphic form is unramified at l.
Chapter 1. Introduction 9
1.3 Jacquet-Langlands correspondence
The Jacquet-Langlands correspondence [24] provides a way to pass from automorphic rep-
resentations of the multiplicative group of a quaternion algebra D′ over a local field to
automorphic representations of the multiplicative group of a different quaternion algebra
D (ramified at a subset of the places where D′ is ramified) — for example from automor-
phic representations of the multiplicative group of a definite quaternion algebra over Q to
automorphic representations of GL2/Q. In this case, the transfer of automorphic repre-
sentations was p-adically interpolated by Chenevier [10] to produce a rigid analytic map
between the eigencurve for the definite quaternion algebra over Q (as constructed in [2])
and the Coleman-Mazur eigencurve for GL2/Q [13].
In Chapter 4 we will be concerned with the Jacquet-Langlands transfer between auto-
morphic representations of the multiplicative group of quaternion algebras over a totally
real field F , which are split at one infinite place (so the attached Shimura varieties are one-
dimensional). These quaternion algebras will always be split at places dividing p. For the
purposes of this introduction we will suppose that F = Q. Let D be a quaternion algebra
over Q, split at the infinite place and with discriminantN , and letD′ be a quaternion algebra
over Q, split at the infinite place and with discriminantNql for q, l distinct primes (coprime
to N ). In this situation there is a rather geometric description of the Jacquet-Langlands
transfer from automorphic representations of (D′⊗Q AQ)× to automorphic representations
of (D ⊗Q AQ)×, arising from the study of the reduction modulo q or l of integral models
of Shimura curves attached to D and D′ (where we consider Shimura curves attached to
D with Iwahori level at q and l). For the case N = 1 and F = Q this is one of the key
results in [38], and it was extended to general totally real fields F by Rajaei [36]. Using
the techniques from these papers, together with Emerton’s completed cohomology, we can
then describe a p-adic Jacquet-Langlands correspondence in a geometric way — the result
obtained is rather stronger than just the existence of a map between eigenvarieties, which
is all that can be deduced from interpolatory techniques as in [10].
1.4 Summary of main results
In this subsection we will summarise the results contained in the following three chapters.
1.4 Summary of main results 10
1.4.1 Level raising
Let D/Q be a definite quaternion algebra. We consider families of finite slope overconver-
gent p-adic automorphic forms as in [2]. We fix an integer N and a prime l, such that p, l
and N are all coprime to each other and the discriminant of D. We use U1(N) and U0(l) to
denote level structures as in Ch.2 §2.2.2. In Ch.2 §2.3 we prove the following result (stated
here in a slightly vague form, see Ch.2 Theorem 2.3.4 for the precise formulation):
Theorem 1.4.2. Let f be an overconvergent eigenform of finite slope and tame levelU1(N),
with Tl eigenvalue tl and Sl eigenvalue sl related by the equation t2l − (l + 1)2sl = 0. The
roots of the lth Hecke polynomial attached to f are equal to α and lα for some α ∈ Cp and
the l-stabilised oldform, with tame level U1(N)∩U0(l), corresponding to the root α lies in
a one-dimensional family of l-new forms.
In Ch.3 Theorem 3.4.2 we prove the same result for families of finite slope overcon-
vergent p-adic modular forms [12], with the added condition that their associated Galois
representations have irreducible reduction mod p. In fact we prove an equivalent formu-
lation of these level raising statements, described in terms of the equidimensionality of a
rigid analytic subspace of the appropriate eigencurve (the ‘l-new’ subspace). The strate-
gies adopted in Ch.2 and Ch.3 are completely different. In Ch.2 we approach the problem
similarly to the way the classical level raising problem for automorphic forms for definite
quaternion algebras is solved in [15, §2]. In Ch.3 we instead use Emerton’s approach to
constructing the eigencurve [19] and prove an equidimensionality result using Emerton’s
Jacquet functor [17] and certain modules over non-commutative Iwasawa algebras.
1.4.3 Level lowering
In Ch.4 we study the completed cohomology of Shimura curves. Using techniques from
[25, 36, 19, 20] we can prove an analogue of Mazur’s principle for p-adic automorphic
forms (see Ch.4 Theorems 4.4.26 and 4.4.29 for precise statements). Roughly speaking,
we show that if a two-dimensional p-adic representation ρ of Gal(F/F ) (F a totally real
field) is unramified at a finite place q - p and occurs in the completed cohomology of a
system of Shimura curves with tame level equal to U0(q) at the q-factor, then the system of
Hecke eigenvalues attached to ρ occurs in the completed cohomology of Shimura curves
with tame level equal to GL2(Oq) at the q-factor.
Chapter 1. Introduction 11
One motivation for this result is its use in applying the machinery of [20] in the special
case where p splits completely in F (so if p is a prime of F dividing p the p-adic local
Langlands correspondence is well defined for representations of GL2(Fp) ∼= GL2(Qp)) —
this should have applications to some cases of the Fontaine-Mazur conjecture for GL2/F
(see also [27]) and the (generalised) weight part of Serre’s conjecture (see [4] and [22]).
1.4.4 A p-adic Jacquet-Langlands correspondence
Finally, also in Ch.4, we show the existence of a short exact sequence of admissible uni-
tary Banach representations of GL2(F ⊗Q Qp) (Ch.4 Theorem 4.4.34 — in fact the exact
sequence is defined integrally) which encodes a p-adic Jacquet-Langlands correspondence
from a quaternion algebra D′/F , split at one infinite place, with discriminant Nq1q2, to a
quaternion algebraD/F , split at the same infinite place and with discriminant N. Applying
Emerton’s eigenvariety construction to this exact sequence gives a map between eigenva-
rieties (Theorem 4.5.18), i.e. an overconvergent Jacquet-Langlands correspondence in the
same spirit as the main theorem of [10]. As a corollary of the results in Ch.4 we also obtain
level raising results (Corollary 4.5.19).
12
Chapter 2
Level raising for p-adic automorphic
forms
Chapter 2. Level raising for p-adic automorphic forms 13
2.1 Introduction
Classical level raising results typically show that if the reduction mod p of a level N modu-
lar form f has certain properties (depending on a prime l 6= p), then there exists a modular
form g of level Nl, new at l, with g ≡ f mod p. An example of a level raising result for
classical modular forms is the following, due to Ribet [37]:
Theorem. Let f ∈ S2(Γ0(N)) be an eigenform, and let p|p be a finite place of Q such that
p ≥ 5 and f is not congruent to an Eisenstein series modulo p. If l - Np is a prime number
such that the following condition is satisfied,
al(f)2 ≡ (1 + l)2 (mod p),
then there exists a l-new eigenform f ∈ S2(Γ0(Nl)) congruent to f modulo p.
In this chapter we prove an analogous level raising result for families of p-adic au-
tomorphic forms. In [2] and Part III of [3], Buzzard defines modules of overconvergent
p-adic automorphic forms for definite quaternion algebras, and constructs from these a so-
called ‘eigencurve’. The eigencurve is a rigid analytic variety whose points correspond to
certain systems of eigenvalues for Hecke algebras acting on these modules of automorphic
forms. This space p-adically interpolates the systems of eigenvalues arising from classical
automorphic forms. Emerton has constructed eigenvarieties in a cohomological framework
[19], but in the following we will work with Buzzard’s more concrete construction. We
have also proved some cases of level raising for p-adic modular forms using the completed
cohomology spaces investigated by Emerton (see Ch.3).
The first construction of an eigencurve was carried out for modular forms (automorphic
forms for GL2) in Coleman and Mazur’s seminal paper [13]. An important recent result
is the construction of a p-adic Jacquet-Langlands map between an eigencurve for a defi-
nite quaternion algebra and the GL2 eigencurve (interpolating the usual Jacquet-Langlands
correspondence), as carried out in [10].
We follow the general approach of the first part of Diamond and Taylor’s paper [15],
and our Theorem 2.2.22 is an analogue of [15, Theorem 1], but several new features appear
in our work. In particular, the level raising results in [44] and [15] for definite quaternion al-
gebras are proved by utilising a pairing on finite dimensional vector spaces of automorphic
forms. In our setting, the spaces of automorphic forms are Banach modules over an affinoid
2.1 Introduction 14
algebra, so we introduce spaces of ‘dual’ automorphic forms and work with the pairing be-
tween the usual space of automorphic forms and the dual space. We then prove suitable
forms of Ihara’s lemma, our Theorem 2.2.19 (cf. lemma 2 of [15]), for the usual and dual
spaces of automorphic forms. An interesting asymmetry between the two situations can be
observed.
This investigation of level raising results was motivated by a conjecture made by A.
Paulin, prompted by results on local-global compatibility on the eigencurve in his thesis
[35]. Paulin’s conjecture was made for the GL2-eigencurve; we may apply our theorem to
the image of the p-adic Jacquet-Langlands map there to prove many cases of his conjecture.
Since we have applications to the eigencurve for GL2/Q in mind we work with definite
quaternion algebras over Q in this chapter, but some of the methods of section 2.2 should
apply to definite quaternion algebras over any totally real number field, although we do use
the fact that weight space is one-dimensional in our arguments. We end this introduction
by stating the conjecture made by Paulin.
2.1.1 A geometric level raising conjecture
We fix two distinct primes p and l, and an integer N coprime to pl. Let E be the cuspidal
Eigencurve of tame level Γ0(Nl), parameterising overconvergent cuspidal p-adic modular
eigenforms (see [3] for its construction). If φ is a point of E , corresponding to an eigen-
form fφ, Paulin defines an associated representation of GL2(Ql), denoted πfφ,l. We call an
irreducible connected component Z of the Eigencurve generically special if the GL2(Ql)-
representations associated to the points of Z away from a discrete set are special. We
define generically unramified principal series similarly. Denote by α and β the roots of the
polynomial X2 − tlX + lsl, where tl and sl are the Tl and Sl eigenvalues of fφ. Paulin
makes the conjecture:
Conjecture. Suppose Z is generically unramified principal series. Suppose further that
there is a point φ on Z where the ratio of α and β becomes l±1 and πfφ,l is special. Then
there exists a generically special component Z ′ intersecting Z at φ.
Chenevier raised the same question (in a slightly different form) in relation to the char-
acterisation of the Zariski closure of the l-new classical forms in the eigencurve. We ad-
dress this issue in section 2.3.5. Finally, in a recent preprint [34] Paulin has proved versions
of his level raising (and lowering) conjectures (even for ramified principal series). His tech-
Chapter 2. Level raising for p-adic automorphic forms 15
niques are completely different to ours, making use of deformation theory and requiring a
recent important result of Emerton [20] showing that the space Xfs constructed by Kisin
in [26] is equal to the GL2-eigencurve (if one restricts to pieces of the two spaces where
certain conditions are satisfied by the relevant mod p Galois representations).
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s
lemma
In this section we will prove the results we need about modules of p-adic overconvergent
automorphic forms for quaternion algebras.
2.2.1 Banach modules
Let K be a finite extension of Qp. We call a normed K-algebra A a Banach algebra if it
satisfies the following properties:
• A is Noetherian,
• the norm | − | is non-Archimedean,
• A is complete with respect to | − |,
• for any x, y in A we have |xy| ≤ |x||y|.
We will normally assume A is a reduced affinoid algebra with its supremum norm. A
Banach A-module is an A-module M endowed with a norm | − | such that
• for any a ∈ A, m ∈M we have |am| ≤ |a||m|,
• M is complete with respect to | − |.
Given a set I we define the Banach A-module cI(A) to be functions f : I → A such that
limi→∞f(i) = 0, with norm the supremum norm. By a finite Banach A-module we mean
a Banach A-module which is finitely-generated as an abstract A-module.
Suppose M is a Banach module over a Banach algebra A. We say that M is ONable
if it is isomorphic (as a Banach module) to some cI(A). Note that this terminology differs
slightly from that of [3], where ONable refers to modules isometric to some cI(A) and
potentially ONable replaces our notion of ONable. The Banach A-module P is said to
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 16
satisfy the universal property (Pr) if for every surjection f : M → N of Banach A-
modules and continuous map α : P → N , α lifts to a continuous map β : P → M such
that the below diagram commutes:
M
f����
P
∃β>>
α // N
As explained in [45, 2.1.4], the universal property (Pr) is the property of being projec-
tive in the category of Banach A-modules, where the notion of projective is defined using
strict epimorphisms of Banach modules (which are just the set-theoretically surjective epi-
morphisms of Banach modules). This is the correct notion of projective object, since the
category of Banach A-modules is an exact category, not an Abelian catefory. A module
P having property (Pr) is equivalent to P being a direct summand of an ONable module.
(See the end of section 2 in [3]).
2.2.2 Some notation and definitions
Let p be a fixed prime. Let D be a definite quaternion algebra over Q with discriminant
δ prime to p. Fix a maximal order OD of D and isomorphisms OD ⊗ Zq ∼= M2(Zq) for
primes q - δ. Note that these induce isomorphisms D ⊗Qq∼= M2(Qq) for q - δ. We define
Df = D ⊗Q Af , where Af denotes the finite adeles over Q. Write Nm for the reduced
norm map from Df to A×f . Note that if g ∈ Df we can regard the p component of g, gp, as
an element of M2(Qp).
For an integer α ≥ 1, we let Mα denote the monoid of matrices
(a b
c d
)∈ M2(Zp)
such that pα|c, p - d and ad− bc 6= 0. If U is an open compact subgroup of D×f and α ≥ 1
we say that U has wild level ≥ pα if the projection of U to GL2(Qp) is contained in Mα.
We will be interested in two key examples of open compact subgroups of D×f . For M
any integer prime to δ, we define U0(M) (respectively U1(M)) to be the subgroup of D×fgiven by the product
∏q Uq, where Uq = (OD⊗Zq)× for primes q|δ, and Uq is the matrices
in GL2(Zp) of the form
(∗ ∗0 ∗
)(respectively
(∗ ∗0 1
)) mod qvalq(M) for all other q. We
can see that if pα divides M , then U1(M) has wild level ≥ pα.
Suppose we have α ≥ 1, U a compact open subgroup of D×f of wild level ≥ pα and A
Chapter 2. Level raising for p-adic automorphic forms 17
a module over a commutative ring R, with an R-linear right action of Mα. We define an
R-module L (U,A) by
L (U,A) = {f : D×f → A : f(dgu) = f(g)up for all d ∈ D×, g ∈ D×f , u ∈ U}
where D× is embedded diagonally in D×f . If we fix a set {di : 1 ≤ i ≤ r} of double coset
representatives for the finite double quotient D×\D×f /U , and write Γi for the finite group
d−1i D×di ∩ U , we have an isomorphism (see section 4 of [2])
L (U,A)→r⊕i=1
AΓi ,
given by sending f to (f(d1), f(d2), . . . , f(dr)). If U ⊂ U1(N) for N ≥ 4, then the groups
Γi are trivial (this is proved in [15]).
For f : D×f → A, x ∈ D×f with xp ∈ Mα, we define f |x : D×f → A by (f |x)(g) =
f(gx−1)xp. Note that we can now also write
L (U,A) = {f : D×\D×f → A : f |u = f for all u ∈ U}.
We can define double coset operators on the spaces L (U,A). If U , V are two compact
open subgroups of D×f of wild level ≥ pα, and A is as above, then for η ∈ D×f with
ηp ∈ Mα we may define an R-module map [UηV ] : L (U,A) → L (V,A) as follows: we
decompose UηV into a finite union of right cosets∐
i Uxi and define
f |[UηV ] =∑i
f |xi.
2.2.3 Overconvergent automorphic forms
Let W be the rigid analytic space Hom(Z×p ,Gm), defined over Qp. The reader may consult
lemma 2 of [2] for details of this space’s construction and properties. For example, W is a
union of finitely many open discs. The space W is the weight space for our automorphic
forms. The Cp-points w of W corresponding to characters κw : Z×p → C×p with κw(x) =
xkεp(x) for some positive integer k and finite order character εp are referred to as classical
weights. Let X be a reduced connected K-affinoid subspace of W , where K/Qp is finite,
and denote the ring of analytic functions on X by O(X). Such a space X corresponds to a
character κ : Z×p → O(X)× induced by the inclusion X ⊂ W . If we have a real number
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 18
r = p−n for some n, then we define Br,K to be the rigid analytic subspace of affine 1-space
over K with Cp-points
Br,K(Cp) = {z ∈ Cp : ∃y ∈ Zp such that |z − y| ≤ r}.
Similarly (for r < 1) we define B×r,K to be the rigid analytic subspace of affine 1-space over
K with Cp-points
B×r,K(Cp) = {z ∈ Cp : ∃y ∈ Z×p such that |z − y| ≤ r}.
A point x ∈ X(Cp) corresponds to a continuous character κx : Z×p → C×p . Such maps are
analytic when restricted to the set {z ∈ Zp : |1 − z| ≤ r} for small enough r. If κx and
r have this property we call x an r-analytic point. A point is r-analytic if and only if its
corresponding character extends to a morphism of rigid analytic varieties
κx : B×r,K → Gm.
Let X be a K-affinoid subspace of W as before, with associated character κ : Z×p →O(X)×. We say that κ is r-analytic if every point inX(Cp) is r-analytic. Fix a real number
0 < r < 1 and let AX,r be the O(X)-Banach algebra O(Br,K ×K X), endowed with the
supremum norm. If κ is rp−α-analytic we can define a right action of Mα on AX,r by, for
f ∈ AX,r, γ =
(a b
c d
)∈Mα,
(f · γ)(x, z) =κx(cz + d)
(cz + d)2f
(x,az + b
cz + d
).
where x ∈ X(Cp) (with κx the associated character) and z ∈ Br,K(Cp).
Definition 2.2.4. Let X be a K-affinoid subspace of W as above, with κ : Z×p → O(X)×
the induced character. If we have a real number r = p−n, some integer α ≥ 1 such
that κ is rp−α-analytic, and U a compact open subgroup of D×f of wild level ≥ pα, then
define the space of r-overconvergent automorphic forms of weight X and level U to be the
O(X)-module
SDX(U ; r) := L (U,AX,r).
Chapter 2. Level raising for p-adic automorphic forms 19
If we endow SDX(U ; r) with the norm |f | = maxg∈D×f|f(g)|, then the isomorphism
SDX(U ; r) ∼=r⊕i=1
A ΓiX,r (2.1)
induced by fixing double coset representatives di is norm preserving. Since the Γi are finite
groups, and AX,r is an ONable Banach O(X)-module (it is the base change to O(X) of
O(Br,K), and all Banach spaces over a discretely valued field are ONable), we see that
SDX(U ; r) is a Banach O(X)-module, and satisfies property (Pr).
Note that if m is a maximal ideal of O(X), corresponding to a point x ∈ X(K ′)
for K ′/K finite, then taking the fibre of the module SDX(U ; r) at m gives the space of
overconvergent forms SDx (U ; r) corresponding to the point x of W (K ′) (note that a point
of W (K ′) is a reduced connected K ′-affinoid subspace!).
These spaces of overconvergent automorphic forms were first defined in [2], using ideas
from the unpublished preprint [43].
2.2.5 Dual modules
Suppose A is a Banach algebra. Given a Banach A-module M we define the dual M∗ to
be the Banach A-module of continuous A-module morphisms from M to A, with the usual
operator norm. We denote the O(X)-module A ∗X,r by DX,r.
If the map κ corresponding to X is rp−α-analytic, then Mα acts continuously on AX,r,
so DX,r has an O(X)-linear right action of the monoid M−1α given by (f · m−1)(x) :=
f(x · m), for f ∈ DX,r, x ∈ AX,r and m ∈ Mα. If U is as in Definition 2.2.4 then its
projection to GL2(Qp) is contained in Mα ∩M−1α , so it acts on DX,r. This allows us to
make the following definition:
Definition 2.2.6. For X , κ, r, α and U as above, we define the space of dual
r-overconvergent automorphic forms of weight X and level U to be the O(X)-module
VDX(U ; r) := L (U,DX,r).
As in 2.2.3, we have a norm preserving isomorphism
VDX(U ; r) ∼=
r⊕i=1
DΓiX,r. (2.2)
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 20
Thus VDX(U ; r) is a BanachO(X)-module. We note that it will not usually satisfy property
(Pr), since (unless X is a point) we expect that DX,r will not be ONable.
If U , V are two compact open subgroups of D×f of wild level ≥ pα, then for η ∈ D×fwith ηp ∈M−1
α we get double coset operators [UηV ] : VDX(U ; r)→ VD
X(V ; r).
2.2.7 Hecke operators
For an integer m, we define the Hecke algebra away from m, T(m), to be the free commuta-
tive O(X)-algebra generated by symbols Tπ, Sπ for π prime not dividing m. If δp divides
m then we can define the usual action of T(m) by double coset operators on SDX(U ; r): for
π - m define $π ∈ Af to be the finite adele which is π at π and 1 at the other places.
Abusing notation slightly, we also write $π for the element of D×f which is
(π 0
0 π
)at π
and the identity elsewhere. Similarly set ηπ =
($π 0
0 1
)to be the element of D×f which is(
π 0
0 1
)at π and the identity elsewhere. On SDX(U ; r) we let Tπ act by [UηπU ] and Sπ by
[U$πU ]. Similarly on VDX(U ; r) we define Tπ to act by [Uη−1
π U ] and Sπ by [U$−1π U ]. As
usual we also have a compact operator acting on SDX(U ; r), namely Up := [UηpU ].
2.2.8 A pairing
In this section X , κ, r, α and U will be as in Definition 2.2.4. We will denote by V
another compact open subgroup of wild level ≥ pα. We fix double coset representatives
{di : 1 ≤ i ≤ r} for the double quotient D×\D×f /U and let γi denote the order of the finite
group d−1i D×di∩U . We can define anO(X)-bilinear pairing between the spaces SDX(U ; r)
and VDX(U ; r) by
〈f, λ〉 :=r∑i=1
γ−1i 〈f(di), λ(di)〉,
where f ∈ SDX(U ; r), λ ∈ VDX(U ; r) and on the right hand side of the above definition 〈·, ·〉
denotes the pairing between AX,r and DX,r given by evaluation.
This pairing is independent of the choice of the double coset representatives di, since
for every d ∈ D×, g ∈ D×f , u ∈ U , f ∈ SDX(U ; r) and λ ∈ VDX(U ; r) we have
〈f(dgu), λ(dgu)〉 = 〈f(g)up, λ(g)up〉 = 〈f(g)upu−1p , λ(g)〉 = 〈f(g), λ(g)〉.
Chapter 2. Level raising for p-adic automorphic forms 21
Combining this observation with the isomorphisms (2.1) and (2.2) we see that our pairing
identifies VDX(U ; r) with SDX(U ; r)∗.
The following proposition summarises a standard computation [44, 15] (although these
assume the level group is small enough that the finite groups Γi are trivial), telling us how
our pairing interacts with double coset operators. In particular, it implies that 〈Tπf, λ〉 =
〈f, Tπλ〉 for π - δp when Tπ acts in the usual way.
Proposition 2.2.9. Let f ∈ SDX(U ; r) and let λ ∈ VDX(V ; r). Let g ∈ D×f with gp ∈ Mα.
Then
〈f |[UgV ], λ〉 = 〈f, λ|[V g−1U ]〉.
Proof. For d ∈ D×f set γ(d) = #(d−1D×d ∩ V ). We have
f |[UgV ] =∑
v∈(g−1Uη)∩V \V
f |(gv),
hence
〈f |[UgV ], λ〉 =∑
d∈D×\D×f /V
γ(d)−1〈f |[UgV ](d), λ(d)〉
=∑
d∈D×\D×f /V
∑v∈(g−1Ug)∩V \V
γ(d)−1〈f |(gv)(d), λ(d)〉
=∑
d∈D×\D×f /V
∑v∈(g−1Ug)∩V \V
γ(d)−1〈f(dv−1g−1) · gpvp, λ(d)〉
=∑
x∈D×\D×f /(g−1Ug)∩V
〈f(xg−1), λ(x) · g−1p 〉
=∑
y∈D×\D×f /U∩(gV g−1)
〈f(y), λ(yg) · g−1p 〉
= 〈f, λ|[V g−1U ]〉
where we pass from the third line to the fourth line by counting double cosets and the final
line follows by similar calculations to the first 5 lines.
By the results of section 5 in [9] and section 3 of [3] we know that for a fixed d ≥ 0, if
X is a sufficiently small affinoid whose norm is multiplicative (with a precise bound given
by Theoreme 5.3.1 of [9]) then since Up acts as a compact operator on SDX(U ; r), we have
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 22
a Up stable decomposition
SDX(U ; r) = SDX(U ; r)≤d ⊕N,
where SDX(U ; r)≤d is the space of forms of slope ≤ d. We need X to be small enough that
the Newton polygon of the characteristic power series for Up acting on SDX(U ; r) has the
same slope ≤ d part when specialised to any point of X .
From now on we fix d and assume that X is such that this slope decomposition exists.
The key example of such an X is an open ball of small radius.
The space SDX(U ; r)≤d is a finite Banach O(X)-module with property (Pr), i.e. a
projective finitely generatedO(X)-module. In fact this decomposition must be stable under
the action of T(δp), since the Tπ and Sπ operators for π 6= p commute with Up. We define
VDX(U ; r)≤d to be the maps from SDX(U ; r) to O(X) which are 0 on N. This space is
also stable under the action of T(δp) and is naturally isomorphic to the dual of SDX(U ; r)≤d.
The following lemma implies that our pairing is perfect when restricted to SDX(U ; r)≤d ×VDX(U ; r)≤d.
Lemma 2.2.10. Let M be a finite Banach O(X)-module with property (Pr). Then the
usual natural map M → (M∗)∗ is an isomorphism. In other words, the O(X)-module M
is reflexive.
Proof. Since M is finite we have a surjection of Banach O(X)-modules O(X)⊕n → M
for some n. Applying the universal property (Pr) to this surjection shows that we have a
Banach O(X)-isomorphism M⊕N ∼= O(X)⊕n for some module N, so M is a projective
O(X)-module. Proposition 2.1 of [3] states that the category of finite Banach O(X)-
modules, with continuous O(X)-linear maps as morphisms, is equivalent to the category
of finiteO(X)-modules so we can just compute duals module-theoretically. We have exact
sequences
0 // M // O(X)⊕n // N // 0
0 // N // O(X)⊕n // M // 0
and since M, N, M∗ and N∗ are all projective as O(X)-modules we can take the dual
Chapter 2. Level raising for p-adic automorphic forms 23
of these exact sequences twice to get commutative diagrams with exact rows
0 // M //
��
O(X)⊕n //
��
N //
��
0
0 // M∗∗ // O(X)⊕n // N∗∗ // 0
0 // N //
��
O(X)⊕n //
��
M //
��
0
0 // N∗∗ // O(X)⊕n // M∗∗ // 0
where the vertical maps are the natural maps from a module to its double dual. Since the
central maps are isomorphisms, we conclude that the outer maps are too.
Direct limits and Frechet spaces
We should remark here that our use of the dual Banach modules VDX(U ; r) is slightly un-
satisfactory. For example, the modules do not satisfy property (Pr), and we must restrict
to ‘slope ≤ d’ subspaces to get a perfect pairing. One could alternatively work with mod-
ules of all overconvergent automorphic forms, rather than imposing r-overconvergence for
a particular r. One defines
SDX(U)† := lim−→r
SDX(U ; r),
where the (compact) transition maps in the direct system are induced by the inclusions
Bs,K ⊂ Br,K for s < r. If X is a point (so O(X) is a field) then it is a standard result that
the vector space SDX(U)† is reflexive (see Proposition 16.10 of [40]). Using this fact it is
fairly straightforward to show that for any X , SDX(U)† is a reflexive O(X)-module, with
dual the Frechet space
VDX(U)† := lim←−
r
VDX(U ; r).
2.2.11 Old and new
Fix an integer N ≥ 1 (the tame level) coprime to p and fix an auxiliary prime l - Npδ.
Let X be an affinoid subspace of weight space with associated character κ which is rp−α
analytic, for some integer α ≥ 1. Set U = U1(Npα), V = U1(Npα) ∩ U0(l). To simplify
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 24
notation we set
L := SDX(U ; r)≤d
L∗ := VDX(U ; r)≤d
M := SDX(V ; r)≤d
M∗ := VDX(V ; r)≤d.
We define a map i : L× L→M by
i(f, g) := f |[U1V ] + g|[UηlV ].
Since the map i is defined by double coset operators with trivial component at p it commutes
with Up and thus gives a well defined map between these spaces of bounded slope forms.
A simple calculation shows that these double coset operators act very simply. Regarding f
and g as functions on D×f we have f |[U1V ] = f , g|[UηlV ] = g|ηl. The image of i inside
M will be referred to as the space of oldforms.
We also define a map i† : M → L× L by
i†(f) := (f |[V 1U ], f |[V η−1l U ]).
The kernel of i† is the space of newforms. The maps i and i† commute with Hecke operators
Tq, Sq, where q - Nplδ.The same double coset operators give maps
j : L∗ × L∗ →M∗,
j† : M∗ → L∗ × L∗.
Using Proposition 2.2.9 we have
〈i(f, g), λ〉 = 〈(f, g), j†λ〉
for f, g ∈ L, λ ∈M∗. Similarly
〈f, j(λ, µ)〉 = 〈i†f, (λ, µ)〉
Chapter 2. Level raising for p-adic automorphic forms 25
for d ∈M , λ, µ ∈ L∗.An easy calculation shows that i†i acts on the product L×L = L2 by the matrix (acting
on the right) (l + 1 [U$−1
l U ][UηlU ]
[UηlU ] l + 1
)=
(l + 1 S−1
l Tl
Tl l + 1
).
We have exactly the same double coset operator formula for the action of j†j on the product
L∗ × L∗ = L∗2. The Hecke operators Sl, Tl act by [U$−1l U ], [Uη−1
l U ] respectively on L∗.
Also, the double coset UηlU is the same as U$lη−1l U , since the matrix which is the identity
at every factor except l and
(0 1
1 0
)at l is in U. From these two facts we deduce that, in
terms of Hecke operators, j†j acts on L∗ × L∗ by the matrix (again acting on the right)(l + 1 Tl
S−1l Tl l + 1
).
If the affinoid X is sufficiently nice, then we can show that the map i†i is injective.
Before we prove this, we note that in our setting a family of p-adic automorphic eigenforms
over an affinoid X ⊂ W is just a Hecke eigenform f in SDX(U ; r).
Proposition 2.2.12. If X is a one dimensional irreducible connected smooth affinoid, then
the map i†i is injective.
Proof. Let L0 be the projective (since O(X) is a Dedekind domain) finite Banach O(X)-
module ker(i†i), and note that L0 ⊂ L2 is stable under the action of all the Hecke operators,
since they all commute with i†i. Suppose L0 is not zero. For (f, g) in L0 we have (l +
1)f + S−1l Tlg = Tlf + (l + 1)g = 0. Eliminating g we get T 2
l f − (l + 1)2Slf = 0,
so projecting L0 down to L (taking either the first or the second factor) we see that the
Hecke operator T 2l − (l + 1)2Sl acts as 0 on a non-zero projective submodule of L. This
(applying the local eigenvariety construction as described in section 6.2 of [9]) implies that
there is a family of eigenforms over some one dimensional sub-affinoid of X , all with the
eigenvalue of T 2l − (l + 1)2Sl equal to 0. Now the Hecke algebra element T 2
l − (l + 1)2Sl
induces a rigid analytic function on the tame level N eigencurve for D (by taking the
appropriate eigenvalue associated to a point), so this function must vanish on the whole
irreducible component containing the one dimensional family constructed above. However,
every irreducible component contains a classical point, and these cannot be contained in the
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 26
kernel of T 2l − (l + 1)2Sl since this would contradict the Hecke eigenvalue bounds given
by the Ramanujan-Petersson conjecture.
Note that the injectivity of i†i implies the injectivity of i. The above shows that if X is
as in the statement of Proposition 2.2.12, we have ker(i†) ∩ im(i) = 0 so our families in
M are not both old and new at l. However, if X is just a point, then i†i may have a kernel -
this corresponds to p-adic automorphic forms which are both old and new at l.
2.2.13 Some modules
We denote the fraction field of O(X) by F . If A is an O(X)-module we write AF for the
F -vector space A⊗O(X) F .
We begin this section by noting that the injectivity of i†i implies the injectivity of j†j:
Suppose j†j(λ, µ) = 0. Then 〈(f, g), j†j(λ, µ)〉 = 0 for all (f, g) ∈ LF , so (by
Proposition 2.2.9) 〈i†i(f, g), (λ, µ)〉 = 0 for all (f, g) ∈ LF . Now since i†i : LF → LF
is an injective endomorphism of a finite dimensional vector space, it is an isomorphism, so
we see that λ = µ = 0. Hence j†j (thus a fortiori j) is injective.
We now define two chains of modules which will prove useful:
Λ0 := L2 Λ∗0 := L∗2
Λ1 := i†M Λ∗1 := j†M∗
Λ2 := i†(M ∩ i(L2F )) Λ∗2 := j†(M∗ ∩ j((L∗F )2))
Λ3 := i†iL2 Λ∗3 := j†jL∗2.
We note that Λ0 ⊃ Λ1 ⊃ Λ2 ⊃ Λ3, and that
Λ2/Λ3 = i†(M ∩ i(L2F )/iL2) = i†((M/iL2)tors),
with analogous statements for the starred modules.
We fix the usual action of T(Nδpl) on all these modules. We can now describe some
pairings between them which will be equivariant under the T(Nδpl) action. They will not all
be equivariant with respect to the action of Tl.
We have a (perfect) pairing 〈, 〉 : L2F × (L∗F )2 → F which, since j is injective, induces
a pairing
Λ0 × (M∗ ∩ j((L∗F )2))→ F/O(X),
Chapter 2. Level raising for p-adic automorphic forms 27
which in turn induces a pairing
P1 : Λ0/Λ1 × (M∗ ∩ j((L∗F )2)/j(L∗2))→ F/O(X).
The fact that this pairing is perfect follows from the following lemma:
Lemma 2.2.14. The pairing on L2F × (L∗F )2 induces isomorphisms
HomO(X)(Λ1,O(X)) ∼= M∗ ∩ j((L∗F )2)
and
HomO(X)(Λ0,O(X)) ∼= j(L∗2).
Proof. For the first isomorphism, elements of the module HomO(X)(Λ1,O(X)) correspond
to l ∈ (L∗F )2 such that 〈i†m, l〉 ∈ O(X) for all m ∈ M . We have 〈i†m, l〉 = 〈m, jl〉 so
〈i†m, l〉 ∈ O(X) for all m ∈ M if and only if 〈m, jl〉 ∈ O(X) for all m ∈ M , i.e. if and
only if jl ∈M∗.
The second isomorphism is obvious, since j is injective.
In exactly the same way, we have a perfect pairing
P2 : (M ∩ i(L2F ))/i(L2)× Λ∗0/Λ
∗1 → F/O(X).
The final pairing we will need is induced by the pairing between M and M∗. It is
straightforward to check that this gives a perfect pairing:
P3 : ker(i†)×M∗/(M∗ ∩ j((L∗F )2))→ O(X).
2.2.15 An analogue of Ihara’s lemma
In classical level raising results (such as [15, 37, 44]) analogues of ‘Ihara’s lemma’ (Lemma
3.2 in [23]) are used to show that prime ideals of a Hecke algebra containing the annihilators
of certain modules of automorphic forms are in some sense ‘uninteresting’, or even to show
that these modules are trivial. In this section we prove the appropriate analogue of Ihara’s
lemma in our setting.
From this section onwards we will assume that X is a one dimensional irreducible con-
nected smooth affinoid in weight space W , so we can apply Proposition 2.2.12. We want
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 28
to obtain information about the T(Nδpl) action on the quotients Λ2/Λ3∼= i†(M/iL2)tors,
Λ∗2/Λ∗3∼= j†(M∗/jL∗2)tors, Λ0/Λ1 and Λ∗0/Λ
∗1. The pairings P1 and P2 allow us to use an
analogue of Ihara’s lemma (the following two propositions and theorem) to obtain crucial
information about all four quotients. Recall that the radius of overconvergence r equals p−n
for some positive integer n. Fix a positive integer c such that Nm(U1(Npα+n)) contains
all elements of Z× congruent to 1 modulo c. We first need a lemma allowing us to control
certain forms with weight a point in weight space.
Lemma 2.2.16. Let x ∈ W (K ′) for some K ′ a finite extension of Qp.
1. Let y ∈ SDx (U ; r) be non-zero. Suppose y factors through Nm, that is y(g) = y(h)
for all g, h ∈ D×f withNm(g) = Nm(h). Then κx is a classical weight z 7→ z2εp(z),
and for all but finitely many primes q ≡ 1 mod c, (where c is the fixed integer chosen
above), (Tq − q − 1)y = 0.
2. Let y ∈ VDx (U ; r). If y factors through Nm, that is y(g) = y(h) for all g, h ∈ D×f
with Nm(g) = Nm(h), then y is zero.
Proof. We first prove part (i). Suppose y is as in the statement of that part. For up ∈
SL2(Qp) ∩ U we have y(g) = y(gup) = y(g) · up for all g ∈ D×f . Noting that
(1 a
0 1
)∈
SL2(Qp) ∩ U for all a ∈ Zp, we see that y(g)(z + a) = y(g)(z) for all a ∈ Zp, z ∈ Br,K′
so y(g)(z) is constant in z, since non-constant rigid analytic functions have discrete zero
sets. Recall that U = U1(Npα), so u0 :=
(1 0
pα 1
)is in SL2(Qp) ∩ U , and for z ∈ Br,K′
we have
y(g)(z) = (y(g)u0)(z) =κx(p
αz + 1)
(pαz + 1)2y(g),
so κx must correspond to the classical weight given by z 7→ z2εp(z) for some character εptrivial on 1 + pα+nZp, where r = p−n. This now implies that for each g ∈ D×f we have
y(g)γ = y(g) for all γ in the projection of U1(Npα+n) to Mα+n, since these matrices all
have bottom right hand entry congruent to 1 mod pα+n.
We now follow [15] to complete the proof of the first part of the lemma. There is a d0 ∈D× with Nm(d0) = q, so Nm(d−1
0 ηq) ∈ A×f is actually in Z× and is congruent to 1 mod c.
Thus (by the way we picked c) there is u0 ∈ U1(Npα+n) such thatNm(u0) = Nm(d−10 ηq).
Now we have
Chapter 2. Level raising for p-adic automorphic forms 29
Tq(y)(g) =∑
u∈(η−1q Uηq)∩U\U
y(gu−1η−1q ) · up
=∑
u∈(η−1q Uηq)∩U\U
y(gη−1q u−1) · up
=∑
u∈(η−1q Uηq)∩U\U
y(gη−1q ) = (q + 1)y(gη−1
q )
= (q + 1)y(gη−1q d0d
−10 ) = (q + 1)y(gu−1
0 d−10 ) = (q + 1)y(gu−1
0 ) = (q + 1)y(g),
where to pass from the first line to the second we use the fact that y factors through Nm
to commute y’s arguments, from the second to the third we use that y is modular of level
U and in the final line we first substitute u0 for d−10 ηq (since they have the same reduced
norm), then commute y’s arguments and use the left invariance of y under D× followed by
the fact that u−10 ∈ U1(Npα+n) implies that y(gu−1
0 ) = y(g)u−10,p = y(g).
We now give a proof of the second part of the lemma. First we perform a formal
calculation. Fix an isomorphism
Ax,r∼=
n∏α=1
K ′〈T 〉
where K ′〈T 〉 is the ring of power series with coefficients in K ′ tending to zero (T a formal
variable), and n is a positive integer depending on r. Such an isomorphism exists since
Br,K′ is just a disjoint union of finitely many affinoid discs. We then have an identification
of Dx,r with∏n
α=1K′〈[T ]〉, where K ′〈[T ]〉 denotes the ring of power series with bounded
coefficients in K ′, and the pairing between Ax,r and Dx,r is given on each component by
〈∑aiT
i,∑bjT
j〉 =∑aibi. Now we can compute the action of γ =
(1 −1
0 1
)on Dx,r.
Let f = (fα)α=1,...,n be an element of Dx,r, with fα =∑bj,αT
j. For each α = 1, ..., n and
i ≥ 0 fix ei,α to be the element of Ax,r which is T i at the α component, and zero elsewhere.
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 30
We have
〈ei,α, f ·
(1 −1
0 1
)〉 = 〈T i, (
∑bj,αT
j) ·
(1 −1
0 1
)〉 = 〈T i ·
(1 1
0 1
),∑
bj,αTj〉
= 〈(T + 1)i,∑
bj,αTj〉
= 〈i∑
k=0
(i
k
)T k,∑
bj,αTj〉
=i∑
j=0
(i
j
)bj,α.
We can now see that if we have f = f ·
(1 −1
0 1
)we get
∑ij=0
(ij
)bj,α = bi,α for all i and
α, which implies that f = 0.
Now we return to the statement in the lemma and suppose y ∈ VDx (U ; r) factors
through Nm. Let u1 be the element of U ⊂ D×f with p component equal to γ and all other
components the identity. For all g ∈ D×f , y(gu1) = y(g)γ since u1 ∈ U and y(gu1) = y(g)
since Nm(u1) = 1. Hence y(g) = y(g)γ and the above calculation shows that y(g) = 0
for all g.
The following two propositions apply the preceding lemma to give a form of Ihara’s
lemma for modules of overconvergent automorphic forms and dual overconvergent forms.
Proposition 2.2.17. For all but finitely many primes q ≡ 1 mod c, Tq − q − 1 annihilates
the module TorO(X)1 (M/iL2,O(X)/m) for each maximal ideal m of O(X).
Proof. Fix q ≡ 1 mod c with q - Npδl and set Hq := Tq − q− 1. Let m be a maximal ideal
ofO(X) and set K ′ = O(X)/m. The maximal ideal m corresponds to a point x of X(K ′),
with corresponding weight κx : Z×p → K ′× the specialisation of κ at m.
We have a short exact sequence
0 // L2 i // M // M/iL2 // 0 .
Noting that L2 and M are O(X)-torsion free, hence flat, and taking derived functors of
Chapter 2. Level raising for p-adic automorphic forms 31
−⊗O(X) K′ gives an exact sequence
0 // TorO(X)1 (M/iL2, K ′)
δ // L2 ⊗O(X) K′
i
||0 M/iL2 ⊗O(X) K
′oo M ⊗O(X) K′oo
.
We have a commutative diagram
0 // L2 i //
Hq
��
M //
Hq
��
M/iL2 //
Hq��
0
0 // L2 i // M // M/iL2 // 0
,
so by the naturality of the long exact sequence for Tor the diagram
TorO(X)1 (M/iL2, K ′)
δ //
Hq��
L2 ⊗O(X) K′
Hq
��
TorO(X)1 (M/iL2, K ′)
δ // L2 ⊗O(X) K′
commutes. To complete the proof it suffices to prove that Hq annihilates the kernel of
i : L2 ⊗O(X) K′ →M ⊗O(X) K
′.
We proceed by viewing these modules as spaces of automorphic forms with weight x (a
single point in weight space). We define two finite dimensional K ′-vector spaces:
Lx := SDx (U ; r)≤d
Mx := SDx (V ; r)≤d.
There are maps ix : L2x → Mx and i†x : Mx → Lx as defined in section 2.2.11 (taking the
weight X in that section to be the point x), but note that as now the weight is just a point
in weight space, Proposition 2.2.12 does not apply. In particular the map ix might not be
injective.
Recall that L and M are finitely generated O(X) modules, whence L ⊗O(X) K′ and
M ⊗O(X) K′ are finite dimensional K ′-vector spaces. Since the Newton polygon of the
characteristic power series forUp acting on SDX(U ; r) has the same slope≤ d part when spe-
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 32
cialised to any point of X , and the specialisation of the Banach O(X)-modules SDX(U ; r)
and SDX(V ; r) at m gives SDx (U ; r) and SDx (V ; r) respectively, we have isomorphisms
L⊗O(X) K′ → Lx and M ⊗O(X) K
′ →Mx which commute suitably with
i : L2 ⊗O(X) K′ → M ⊗O(X) K
′ and ix : L2x → Mx. These isomorphisms also commute
with double coset operators, so to prove the proposition it suffices show that Hq annihilates
the kernel of ix.
Suppose ix(y1, y2) = 0. Then y1 = −y2|ηl, so we have y2 ∈ SDx (U ; r), y2|ηl ∈SDx (U ; r). Therefore y2 and y2|ηl are both invariant under the action of the group U , so y2
is invariant under the action of the group generated by U and ηlUη−1l in D×f . (Note that
every element of this group has projection to its pth component lying in Mα.) Since by
II.1.4, Corollary 1 of [42]
SL2(Ql) = 〈SL2(Zl),
(l 0
0 1
)SL2(Zl)
(l 0
0 1
)−1
〉,
and the l-factor of U is GL2(Zl), we have that y2 is invariant under SL2(Ql), where we
embed SL2(Ql) into D×f in the obvious way.
Denote by DNm=1 the algebraic subgroup of D× whose elements are of reduced norm
1. We haveDNm=1(Ql) ∼= SL2(Ql), sinceD is split at l. The strong approximation theorem
applied to DNm=1 implies that DNm=1(Q) · SL2(Ql) is dense in DNm=1f := DNm=1(Af ),
where DNm=1(Q) is embedded diagonally in DNm=1f . For each g ∈ D×f we define
Xg := {h ∈ DNm=1f : y2(gh) = y2(g)}.
Since y2 is continuous, Xg is closed, and for δ ∈ DNm=1(Q), γ ∈ SL2(Ql) we have
y2(gg−1δγg) = y2(δγg) = y2(γg) = y2(gg−1γg) = y2(g), since g−1γg ∈ SL2(Ql). There-
fore Xg contains the dense set g−1DNm=1(Q)SL2(Ql)g, so Xg is the whole of DNm=1f .
This shows that y2 factors through Nm. Now the first part of Lemma 2.2.16 applies.
Proposition 2.2.18. The module TorO(X)1 (M∗/jL∗2,O(X)/m) is 0 for all maximal ideals
m of O(X)
Proof. We again set K ′ = O(X)/m, and let x ∈ X(K ′) be the point corresponding to m.
Proceeding as at the beginning of the proof of Proposition 2.2.17 we see that we must show
that the map
j : L∗2 ⊗O(X) K′ →M∗ ⊗O(X) K
′
Chapter 2. Level raising for p-adic automorphic forms 33
is injective. We define
L∗x := VDx (U ; r)≤d
M∗x := VD
x (V ; r)≤d.
As in the previous proposition, it is sufficient to show that the map jx : L∗2x → M∗x is
injective. Now we continue as in the proof of Proposition 2.2.17, and finally apply the
second part of Lemma 2.2.16.
The following consequence of the preceding two propositions will be the most conve-
nient analogue of Ihara’s lemma for our applications.
Theorem 2.2.19. 1. There is a positive integer e such that for all but finitely many
primes q ≡ 1 mod c, (Tq − q − 1)e annihilates (M/iL2)tors. Therefore these Hecke
operators annihilate the modules Λ2/Λ3 and, by consideration of the pairing P2,
Λ∗0/Λ∗1.
2. The module (M∗/jL∗2)tors is equal to 0. Therefore the modules Λ∗2/Λ∗3 and, by con-
sideration of the pairing P1, Λ0/Λ1 are also equal to 0.
Proof. We first prove the first part of the theorem. Fix a q ≡ 1 mod c with q - Npδl. The
module (M/iL2)tors is finitely generated (and torsion) over the Dedekind domain O(X),
so it is isomorphic as an O(X)-module to⊕
iO(X)/meii for some finite set of maximal
ideals mi in O(X). We set e to be the maximum of the ei. Set Hq := Tq − q − 1 as before.
We will show that Heq annihilates (M/iL2)tors.
Indeed, suppose that m ∈ M represents a nonzero torsion class in M/iL2. Thus, there
exists a nonzero α ∈ O(X) such that αm ∈ iL2. We have
TorO(X)1 (M/iL2,O(X)/(α)) = {m ∈M/iL2 : αm = 0}.
So we are required to prove that Heq annihilates
TorO(X)1 (M/iL2,O(X)/(α)).
Since
(M/iL2)tors ∼=⊕i
O(X)/meii ,
2.2 Modules of p-adic overconvergent automorphic forms and Ihara’s lemma 34
we can assume that (α) ⊃∏
imeii , so it is enough to show that He
q annihilates
⊕i
TorO(X)1 (M/iL2,O(X)/mei
i ).
Taking derived functors of −⊗O(X) M/iL2 of the short exact sequence
0 // mi/meii
// O(X)/meii
// O(X)/mi// 0
and applying Proposition 2.2.17 and induction on ei (note that mi/meii is isomorphic to
O(X)/mei−1i ), we see that for each i, Hei
q annihilates TorO(X)1 (M/iL2,O(X)/mei
i ). Now
ei ≤ e for all i, so Heq annihilates all of Tor
O(X)1 (M/iL2,O(X)/(α)). Since α was arbi-
trary, Heq annihilates (M/iL2)tors.
The second part of the theorem follows easily from Proposition 2.2.18.
2.2.20 Supporting Hecke ideals
We will call a maximal ideal M in the Hecke algebra T(Nδp) Eisenstein (compare [30]) if
there is some positive integer c such that for all but finitely many primes q ≡ 1 mod c,
Tq − q − 1 ∈M. The motivation for this definition is the following well-known lemma:
Lemma 2.2.21. Suppose we have a Galois representation ρ : Gal(Q/Q) → GL2(Qp),
continuous and unramified at all but finitely many primes, and a positive integer c such
that for all but finitely many primes q ≡ 1 mod c, ρ is unramified at q and the trace of
Frobenius at q, Tr(Frobq) = q + 1. Then ρ is reducible.
Proof. The formula for the traces implies that ρ restricted to the absolute Galois group
of the cyclotomic field Q(ζc) has semisimplification isomorphic to 1 ⊕ χ, where 1 is the
trivial one dimensional representation, and χ is the p-adic cyclotomic character. Denoting
Gal(Q/Q) by G and Gal(Q/Q(ζc)) by H and applying Frobenius reciprocity we conclude
that
Hom(IndGH(1⊕ χ), ρ)
is non-zero. IndGH(1 ⊕ χ) is just a direct sum of one dimensional representations, so ρ is
reducible.
We recall that for an arbitrary (commutative) ring R the support of an R-module A
is the set of prime ideals p C R such that the localisation Ap is non-zero. If A is finitely
Chapter 2. Level raising for p-adic automorphic forms 35
generated as an R-module then the support of A is equal to the set of prime ideals in R
containing the annihilator of A. We write TL for the image of T(Nδp) in EndO(X)(L) and
similarly TM for the image of T(Nδpl) in EndO(X)(M). Analogously we define TL∗ and
TM∗ to be the image of the Hecke algebra in the endomorphism rings of the relevant dual
modules. Note that there are natural maps TM → TL and TM∗ → TL∗ . If I is an ideal of
T(Nδp) we write IL for the image of I in TL and I ′M for the image of I ∩ T(Nδpl) in TMWe can now state and prove the main theorem of this section.
Theorem 2.2.22. Suppose M is a non-Eisenstein maximal ideal of T(Nδp) containing T 2l −
(l+ 1)2Sl. Further suppose that ML is in the support of the TL-module L. Then M′M is in
the support of the TM -module ker(i†) ⊂M .
Proof. We write ML∗ for the image of M in TL∗ and M′M∗ for the image of M ∩ T(Nδpl)
in TM∗ . Since ML is in the support of L, and the perfect pairing between L and L∗ is
equivariant with respect to all of T(Nδp) (including Tl), we know that ML∗ is in the support
of L∗. Consider the module
Q := Λ∗0/Λ∗3 = L∗2
/L∗2
(l + 1 Tl
S−1l Tl l + 1
),
Since L∗2ML∗6= 0 and det
(l + 1 Tl
S−1l Tl l + 1
)∈ M we know that QML∗ 6= 0, i.e. ML∗ is in
the support of Q. We can view Q as a TM∗-module, with M′M∗ in its support.
Theorem 2.2.19 implies that if M′M∗ is in the support of Λ∗0/Λ
∗1 or Λ∗2/Λ
∗3 then it is
Eisenstein, so it must be in the support of Λ∗1/Λ∗2. This quotient is a homomorphic image
of M∗/(M∗ ∩ j(L∗2F )), so M′M∗ is in the support of M∗/(M∗ ∩ j(L∗2F )). Finally we can
apply pairing P3 (which is equivariant with respect to T(Nδpl)) to conclude that M′M is in
the support of ker(i†).
2.3 Applications
In this section we explain some applications of the preceding results, including the proof
of some cases of the conjecture mentioned in the introduction.
2.3 Applications 36
2.3.1 Geometric level raising for p-adic modular forms
We firstly describe the application of Theorem 2.2.22 to the conjecture of our Introduction.
Chenevier [10] extended the classical Jacquet-Langlands correspondence to a rigid analytic
embedding from the eigencurve for a definite quaternion algebra to some part of the GL2
eigencurve. We may use this to translate the results of the previous section to the GL2
eigencurve.
We state the case of Theorem 3 in [10] that we will use. We have primes p, l and a
coprime integer N ≥ 1. Fix another prime q and a character ε of Z/NpZ. Let E be the
tame level Γ1(N) ∩ Γ0(q) and character ε reduced cuspidal eigencurve. Let D/Q be a
quaternion algebra ramified at the infinite place and at q, and let E D be the corresponding
reduced eigencurve of tame level U1(N) and character ε.
Theorem 2.3.2. There is a closed rigid analytic immersion
JLp : E D ↪→ E
whose image is the Zariski closure of the classical points of E that are new at q. This map
is defined over weight space and is Hecke equivariant.
We get the same result if we change the level of E to Γ1(N)∩Γ0(ql) (call this eigencurve
E ′) and change the level of E D to U1(N) ∩ U0(l) (call this eigencurve E D′), where we
construct these eigencurves using the Hecke operators at l in addition to the usual Hecke
operators away from the level. This allows us to relate E ′ and the two-covering E old of E
corresponding to taking roots of lth Hecke polynomial.
Lemma 2.3.3. There is a closed embedding E old ↪→ E ′, with image the Zariski closure of
the classical l-old points in E ′.
Proof. Given X an affinoid subdomain of W , let MX and M ′X denote the Banach O(X)-
modules of families (weight varying over X) of overconvergent modular forms of tame
levels Γ1(N) ∩ Γ0(q) and Γ1(N) ∩ Γ0(ql) respectively, as defined in section 7 of [3]. The
two degeneracy maps from level Γ1(N) ∩ Γ0(ql) to level Γ1(N) ∩ Γ0(q) give a natural
embedding M2X → M ′
X . Denote the image of this map by MoldX - it is stable under all
Hecke operators (including at l). The lemma follows from the observation that applying
the eigenvariety machine of [3] to the Banach modules MoldX (with X varying) gives the
space E old.
Chapter 2. Level raising for p-adic automorphic forms 37
Let Z ⊂ E ′ be the Zariski closure of the classical points in E ′ corresponding to forms
new at l. Proposition 4.7 of [10] shows that Z can be identified with the points of E ′
lying in a one dimensional family of l-new points, where l-new means they come from
overconvergent modular forms in the kernel of the map analogous to i† in the GL2 setting.
The following theorem corresponds to the conjecture in the introduction for points of E ′ in
the image of JLp.
Theorem 2.3.4. Suppose we have a point φ ∈ E lying in the image of JLp, with T 2l (φ) −
(l + 1)2Sl(φ) = 0. Let the roots of the lth Hecke polynomial corresponding to φ be α and
lα where α ∈ Cp. Then the point over φ of E old corresponding to α lies in Z .
Proof. We pick d, r and α such that the automorphic form corresponding to the preimage
of φ under JLp is r-overconvergent of slope ≤ d and level U1(Npα). Now fix a closed ball
in W , containing the weight of φ, which is small enough (note that ‘small enough’ depends
on d, r and α) to apply the local eigenvariety construction described in section 6.2 of [9].
Denote this ball by X . As usual the rp−α-analytic character Z×p → O(X) induced by the
embedding X ↪→ W is denoted by κ.
The system of Hecke eigenvalues given by φ corresponds to a maximal ideal M in
the O(X) Hecke algebra T(Npq). We know that T 2l − (l + 1)2Sl ∈ M. If we set L =
SDX(U1(Npα); r)≤d as before, and use the notation of the previous section, then ML is
in the support of L. At this stage Theorem 2.2.22 applies to the ideal M, which is not
Eisenstein since the Galois representation attached to φ is irreducible (recall that we are
working on the cuspidal part of the eigencurve). Therefore we know that M′M is in the
support of ker(i†) ⊂M . We can then take a height one prime ideal p ⊂M′M in the support
of ker(i†) ⊂M , and then this corresponds (by proposition 6.2.4 of [9]) to a p-adic family of
automorphic forms, new at l, passing through a point φ′ with system of Hecke eigenvalues
the same as those for φ away from l. Now applying the map JLp we see that one of the
points over φ must lie in Z . A calculation using the fact that φ′ comes from an eigenform
in the kernel of i† shows that the point corresponds to the root α.
To translate this theorem into the language of the introduction, note that E old corre-
sponds to the generically unramified principal series components of E ′, whilst Z corre-
sponds to the generically special or supercuspidal components. We may identify the point
over φ lying in Z as the one whose attached GL2(Ql)-representation is special.
2.3 Applications 38
2.3.5 Eigenvarieties of newforms
We return to the situation of a definite quaternion algebra D over Q with arbitrary discrim-
inant δ prime to p. Denote the levels U1(Npα) by U and U1(Npα)∩U0(l) by V , as before.
We denote by E D the tame level U1(N) ∩ U0(l) reduced eigencurve for D. Suppose φ is a
point of E D, with weight x. We say that φ is l-new if it corresponds to a Hecke eigenform in
the kernel of the map i†x : SDx (V ; r)→ SDx (U ; r), where this is defined as in section 2.2.11
by i†x(f) := (f |[V 1U ], f |[V η−1l U ]). Denote by Z the Zariski closure of the points in E D
arising from classical l-new forms. We have the following proposition, due to Chenevier.
Proposition 2.3.6. 1. The set of x in E D that are l-new is the set of points of a closed
reduced analytic subspace E Dnew ⊂ E D.
2. Z is a closed subspace of E Dnew, and its complement is the union of irreducible com-
ponents of dimension 0 in E Dnew.
3. A point of E Dnew lies in Z if and only if it lies in a one-dimensional family of points in
E Dnew.
Proof. Exactly as for Proposition 4.7 of [10].
We now apply the results of section 2.2 to show that a point of E Dnew\Z lies in a one-
dimensional family of points in E Dnew, so by contradiction we can conclude that E D
new is
equal to Z .
Theorem 2.3.7. E Dnew is equal to Z . In particular, E D
new is equidimensional of dimension 1.
Proof. Let φ be a point of E Dnew\Z with weight x. The proof will proceed by showing that
φ is also l-old, then raising the level at φ, as in the previous theorem, to show that it lies in
a family of l-new points.
We pick d, r and α such that φ comes from a r-overconvergent automorphic form of
slope d and level U = U1(Npα). Now fix a closed ball in W , containing the weight of φ,
which is small enough (note that ‘small enough’ depends on d, r and α) to apply the local
eigenvariety construction described in section 6.2 of [9]. Denote this ball by X . The point
x in X corresponds to a maximal ideal m of O(X). If we set M = SDX(V ; r)≤d as before,
then φ lies in a family corresponding to a Hecke eigenvector in M .
As in the previous subsection, we have a closed embedding E Dold ↪→ E D, where E D
old
is a two-covering of the tame level U reduced eigencurve for D, and the image of this
Chapter 2. Level raising for p-adic automorphic forms 39
embedding is the Zariski closure of the classical l-old points in E D, which also equals the
space of l-old points in E D (as is clear from applying the proof of Lemma 2.3.3 to modules
of overconvergent automorphic forms for D). Since the space E D is equidimensional of
dimension 1, but φ does not lie in a family of points in E Dnew, φ must lie in a family of points
in E Dold. So φ is l-old and l-new.
We now ‘raise the level’ at φ. As in the proof of Proposition 2.2.17 we specialise the
O(X) modules L = SDX(U ; r)≤d and M at the maximal ideal m to give vector spaces Lxand Mx. Since φ is l-new and l-old, it arises from an eigenform g in im(ix) ∩ ker(i†x).
Now a calculation using the explicit matrix for the map i†xix shows that g is of the form
ix(αf,−f), where f is an eigenform in Lx with (T 2l − (l + 1)2Sl)f = 0, and the roots
of the lth Hecke polynomial for f are α and lα (note that with our normalisations g is the
l-stabilisation of f corresponding to the root α). Now applying the proof of Theorem 2.3.4
we see that φ lies in Z , so we have a contradiction and therefore must have E Dnew = Z .
Modules of newforms and the eigenvariety machine
Define the Banach O(X)-module MnewX,r for varying affinoid subdomains X ⊂ W (with
corresponding character κ) to be the kernel of the map
i† : SDX(U1(Npα) ∩ U0(l); r)→ SDX(U1(Npα); r)× SDX(U1(Npα); r).
One might wish to construct the rigid analytic space E Dnew from these modules using Buz-
zard’s eigenvariety machine [3]. The key issue is to show that the modules MnewX,r behave
well under base change between affinoid subdomains. The author is not sure whether this
should be true or not - the results in this chapter can be viewed as showing that these mod-
ules behave well under base change from a sufficiently small affinoid to a point and it is
not clear that one can conclude something about base change between open affinoids from
this.
40
Chapter 3
Completed cohomology and level raising
for p-adic modular forms
Chapter 3. Completed cohomology and level raising for p-adic modular forms 41
3.1 Introduction
In this note we describe an application of Poincare duality for completed homology spaces
(as defined by Emerton) to level raising for p-adic modular forms. Fix a prime l 6= p and
a positive integer N coprime to pl. Let D(N) and D(N, l) be the reduced eigencurves of
tame level Γ1(N) and Γ1(N) ∩ Γ0(l) respectively, constructed using the Hecke operators
away from N (we do use the Hecke operators at l). Let D(N, l)c be the equidimensional
closed rigid analytic subvariety ofD(N, l) whose points are those with irreducible attached
residual Galois representation ρ : Gal(Q/Q) → GL2(Fp), and denote by D(N)c the anal-
ogous subvariety of D(N). Define D(N, l)newc to be the the closed rigid analytic subvariety
of D(N, l)c whose points come from overconvergent modular forms which vanish under
the two natural trace maps from forms of level Γ1(N)∩Γ0(l) to forms of level Γ1(N). Our
main result is the following, proved in section 3.4:
Theorem. D(N, l)newc is equidimensional of dimension 1. It is equal to the Zariski closure
of the classical l-new points in D(N, l)c.
This answers a question raised by the main theorem of [10], where the image of a p-adic
Jacquet-Langlands map is identified as the Zariski closure of the classical l-new points in
D(N, l). At least in the subspace D(N, l)c, the above theorem allows a point in this image
to be identified by a natural condition on the corresponding overconvergent modular form.
This theorem is equivalent to a level raising result on the eigencurve. For each point
x of D(N)c there are 2 points x1, x2 of D(N, l)c corresponding to the two roots of the
lth Hecke polynomial for x. Suppose T 2l (x) − (l + 1)2Sl(x) = 0. Equivalently, one of
the two points x1, x2, say x1, is in D(N, l)newc . The above theorem implies that there is a
1-dimensional irreducible component of D(N, l)newc passing through x1. This irreducible
component is an irreducible component of D(N, l)c, with the property that every classical
point on it is l-new. The point x1 is not smooth, since two irreducible components (an old
one and a new one) intersect there.
This short chapter builds on Ch.2 above, where an analogous theorem is proved for the
eigencurve corresponding to overconvergent automorphic forms on a definite quaternion
algebra - in fact in that setting our results include the points where the attached residual
Galois representation is reducible. This difference arises because in Ch.2 a form of Ihara’s
lemma in characteristic 0 is used, whereas in this chapter we make use of the classical
Ihara’s lemma to show the injectivity of a level raising map between completed cohomol-
3.2 Completed cohomology 42
ogy spaces. We make use of completed cohomology (and homology) because Poincare
duality provides us with a natural pairing between completed homology spaces, which
are finitely generated modules under a non-commutative local ring. Constructing a suit-
able pairing between spaces of overconvergent modular forms, the approach taken in Ch.2,
seems to be rather more difficult in the more geometric setting of overconvergent modular
forms.
3.2 Completed cohomology
We begin by defining completed homology and cohomology spaces, following Emerton’s
seminal paper [19] and Calegari and Emerton’s survey [5]. Fix a finite extension E of
Qp, with ring of integers O and uniformiser $. Let Kp be a compact open subgroup
of GL2(Apf ). If Kp is a compact open subgroup of G := GL2(Qp) then we denote by
Y (KpKp) the open modular curve (over C) of level KpK
p and write
H i(Kp, A) := lim−→Kp
H i(Y (KpKp), A),
where the limit is over all compact open Kp, and A is either E, O or O/$s for some
s > 0. Note that there is a natural G action on H i(Kp, A), and a natural G-equivariant
isomorphism
H i(Kp,O)/$sH i(Kp,O) ∼= H i(Kp,O/$s).
Definition 3.2.1. We define the completed ith cohomology space of tame level Kp to be
H i(Kp,O) := lim←−s
H i(Kp,O)/$sH i(Kp,O) ∼= lim←−s
H i(Kp,O/$s).
Applying the analogue of this construction to compactly supported cohomology, we
obtain the space H ic(K
p,O). We can also form completed homology spaces in a dual man-
ner:
Definition 3.2.2. The completed ith homology space of tame level Kp is
Hi(Kp,O) := lim←−
Kp
Hi(Y (KpKp),O).
The same construction with Borel-Moore homology gives HBMi (Kp,O). We write G0
Chapter 3. Completed cohomology and level raising for p-adic modular forms 43
for GL2(Zp) ⊂ G. The key finiteness property for the spaces we have defined above is
contained in the following result:
Proposition 3.2.3. 1. Hi(Kp,O) and HBM
i (Kp,O) are finitely generated left modules
under the natural action of O[[G0]]. The topology induced by the O[[G0]]-module
structure is equivalent to the projective limit topology on these spaces.
2. We have Homcts(HBM1 (Kp,O),O) ∼= H1
c (Kp,O) and Homcts(H1c (Kp,O),O) ∼=
HBM1 (Kp,O).
Proof. The first part of the proposition follows from the first part of Theorem 1.1 in [5]. As
for the second part, Proposition 4.3.6 of [19] shows that H2c (Kp,O) = 0, whilst viewing
Borel-Moore homology as the homology of the compactified modular curve relative to the
cusps, it is easy to see that HBM0 (Kp,O) = 0. Our proposition now follows from the third
part of Theorem 1.1 in [5].
Remark. The statement of Theorem 1.1 in [5] also describes the connection between
Hi(Kp,O) and H i(Kp,O), which is slightly more complicated than in the compactly sup-
ported case, since H0(Kp,O) is non-zero.
Definition 3.2.4. The Hecke algebra T(Kp) is defined to be the weakly closedO-algebra of
G-equivariant endomorphisms of H1(Kp,O) topologically generated by Hecke operators
Tq, Sq for primes q - p that are unramified in Kp.
3.3 Non-optimal levels
We now work with two fixed tame levels,
K(N) := {g ∈ GL2(Zp) : g ≡
(∗ ∗0 1
)mod N}
and
K(N, l) := K(N) ∩ {g ∈ GL2(Zp) : g ≡
(∗ ∗0 ∗
)mod l}.
We have a continuous morphism T(K(N, l)) → T(K(N)), with image T(K(N))(l) the
subalgebra of T(K(N)) generated by the Hecke operators away from l. The following
definition can be found in section 1.2 of [1].
3.3 Non-optimal levels 44
Definition 3.3.1. An ideal I of T(K(N))(l) is Eisenstein if the map λ : T (K(N))(l) →T (K(N))(l)/I satisfies λ(Tq) = ε1(q) + ε2(q) and λ(qSq) = ε1(q)ε2(q) for all q - plN , for
some characters ε1, ε2 : Z×plN → T (K(N))(l)/I .
Fix a maximal ideal m of T(K(N))(l) which is not Eisenstein. We can pull back m to a
maximal ideal m of T(K(N, l)) (maximal since m will certainly contain p).
Proposition 3.3.2. The natural maps α : H1c (K(N),O)→ H1(K(N),O) and
β : H1c (K(N, l),O)→ H1(K(N, l),O) become isomorphisms after localising at the non-
Eisenstein maximal ideals m and m respectively. The same holds for the maps from usual
to Borel-Moore homology.
Proof. Proposition 4.3.9 of [19] shows that the maps α and β are surjective. Let M denote
the kernel of α. Note that M is a torsion free O-module, since H1c (K(N),O) is a torsion
freeO-module. Corollaire 3.1.3 of [1] shows that M is ‘Eisenstein’, i.e. if there is a system
of Hecke eigenvalues λ : T (K(N))(l) → L, L a finite extension of E, with MλL 6= 0, then
ker(λ) is an Eisenstein ideal.
Note that M contains a dense subspace M := ker(H1c (K(N),O) → H1(K(N),O)).
In fact M is equal to the $-adic completion of M , one way to see this is using the explicit
description of M (and M ) given by Proposition 3.1.1 of [1]. The space M⊗OE is spanned
by Hecke eigenvectors, which have Eisenstein systems of eigenvalues, so (M ⊗O E)m = 0
and henceMm = 0, sinceM isO-torsion free. Now Mm is a direct summand of M , and the
projection M → Mm is $-adically continuous, with M in the kernel, hence this projection
is the zero map and Mm = 0. So α does give an isomorphism after localising at m.
The same argument applies to the map β. The statement for homology follows from
the duality of the second part of Proposition 3.2.3 (note that this duality holds for usual
homology and cohomology after localising at a non-Eisenstein maximal ideal, since this
kills H0(Kp,O) by the same argument we used to show that Mm = 0).
Remark. In the proof of Proposition 7.7.13 of [18] the fact that localising at non-Eisenstein
maximal ideals induces an isomorphism between compactly supported completed cohomol-
ogy and completed cohomology is used. The author thanks Matthew Emerton for commu-
nicating the above argument for why this follows from Corollaire 3.1.3 of [1].
Since the Hecke algebras are semilocal, localising at the maximal ideals m and m
gives us direct summands of the original (co)homology spaces. Note that the second
Chapter 3. Completed cohomology and level raising for p-adic modular forms 45
part of Proposition 3.2.3 now gives us dualities between H1(K(N),O)m (respectively
H1(K(N, l),O)m) and H1(K(N),O)m (respectively H1(K(N, l),O)m).
For each compact open Kp ⊂ G there are two degeneracy maps from Y (KpK(N, l))
to Y (KpK(N)), which in the limit give rise to a natural level raising map
i : H1(K(N),O)2m → H1(K(N, l),O)m.
There is a map in the other direction
i† : H1(K(N, l),O)m → H1(K(N),O)2m
given by the inverse system of maps
i†s : H1c (KpK(N, l),O/$s)→ H1
c (KpK(N),O/$s)2
where i†s is the adjoint under Poicare duality of the usual level raising map (as described in
section 3 of [15] for the Shimura curve case). We can form the O-duals of these maps to
get maps i∗, i†∗ between homology spaces.
A standard calculation shows that the composition i† · i acts by the matrix(l + 1 Tl
S−1l Tl l + 1
)
on H1(K(N),O)2m.
Definition 3.3.3. We define the l-new space to be
H1new(K(N, l),O)m := ker(i†).
By duality, we can identify the dual Homcts(H1new(K(N, l),O)m,O) with
Hnew1 (K(N, l),O)m := H1(K(N, l),O)m/i
†∗(H1(K(N),O)2m).
There is a form of Ihara’s lemma in this situation, which follows easily from the classi-
cal Ihara’s lemma:
Lemma 3.3.4. The map i is an injection. Dually, the map i∗ is a surjection.
3.3 Non-optimal levels 46
Proof. It is enough to show that the induced map
i : H1(K(N),O)2m/$
s → H1(K(N, l),O)m/$s
is an injection for all s. Let T be the abstract Hecke algebra over O generated by Tq, Sq for
primes q - plN . It is clear that m and m pull back to the same non-Eisenstein maximal ideal
M of T. Since H1(K(N),O)m/$s ∼= H1(K(N),O)M/$
s and H1(K(N, l),O)m/$s ∼=
H1(K(N, l),O)M/$s we just need to show that the map
i : H1(K(N),O)2M → H1(K(N, l),O)M
is injective with torsion free cokernel, which is the usual Ihara’s lemma as in [37].
Poincare duality at finite levels also gives rise to a duality between completed homology
spaces:
Proposition 3.3.5. There are O[[Kp]]-module isomorphisms
δ1 : Hom(H1(K(N),O)m,O[[Kp]]) ∼= H1(K(N),O)m
δ2 : Hom(H1(K(N, l),O)m,O[[Kp]]) ∼= H1(K(N, l),O)m,
where Kp is any open subgroup of G0.
Proof. This follows from localising the Poincare duality spectral sequence that can be
found in [5].
Lemma 3.3.6. We have a commutative diagram
Hom(H1(K(N),O)2m,O[[Kp]])
t(i∗) //
∼ δ1⊕δ1��
Hom(H1(K(N, l),O)m,O[[Kp]])
∼ δ2��
H1(K(N),O)2m
i†∗ // H1(K(N, l),O)m
,
where the map t(i∗) is the O[[Kp]]-dual of i∗.
Proof. It suffices to check that for each K ′p an open normal subgroup of Kp, the mapinduced by t(i∗) on the finite level quotient H1(K(N)K ′p,O)2
m is equal to the map inducedby i†∗. This follows from the adjointness of i and i† under classical Poincare duality, and
Chapter 3. Completed cohomology and level raising for p-adic modular forms 47
the fact that the duality of Proposition 3.3.5 is induced by the maps
HomO[Kp/K′p](H1(K(N)K ′
p,O),O[Kp/K′p]) ∼= HomO(H1(K(N)K ′
p,O),O)→ HBM1 (K(N)K ′
p,O),
where the first map is a canonical isomorphism of O[Kp/K′p]-modules (as in Lemma 6.1
of [16]) and the second map is given by classical Poincare duality.
Say that a compact open subgroup Kf of GL2(Af ) is neat if GL2(Q) acts on (C\R)×GL2(Af )/Kf without fixed points. The following proposition is due to Emerton [20, Corol-
lary 5.3.19], see also Proposition 4.5.6 and Corollary 4.5.8 below for the proof of a similar
result:
Proposition 3.3.7. For a compact open Kp ⊂ G which is pro-p, such that KpK(N) and
KpK(N, l) are neat (equivalentlyKpK(N) is neat), H1(K(N),O)m and H1(K(N, l),O)m
are free O[[Kp]]-modules.
Theorem 3.3.8. If Kp satisfies the conditions of the above proposition, then
Hnew1 (K(N, l),O)m is also a free O[[Kp]]-module.
Proof. By lemma 3.3.4, there is a short exact sequence
0 // ker(i∗) // H1(K(N, l),O)mi∗ // H1(K(N),O)2
m// 0 .
Since H1(K(N),O)2m is free, there is a section to i∗, and so ker(i∗) is a direct summand of
the free module H1(K(N, l),O)m. Hence ker(i∗) is projective. But we assumed Kp pro-p, so O[[Kp]] is local implying (by the noncommutative version of Nakayama’s lemma)that ker(i∗) is free. Now applying the functor Hom(−,O[[Kp]]) to the above short exactsequence of free modules, we get a diagram
0 −→ Hom(H1(K(N),O)2m,O[[Kp]]) −→ Hom(H1(K(N, l),O)m,O[[Kp]]) −→ Hom(ker(i∗),O[[Kp]]) −→ 0
∼y ∼
yH1(K(N),O)2m
i†∗−−→ H1(K(N, l),O)m,
where the first row is short exact, the vertical maps are the isomorphisms provided by
Proposition 3.3.5 and the horizontal map on the second row is identified as i†∗ by Lemma
3.3.6. Hence this diagram gives an isomorphism of O[[Kp]]-modules between the free
module Hom(ker(i∗),O[[Kp]]) and coker(i†∗) = Hnew1 (K(N, l),O)m.
3.4 Eigencurves of newforms 48
3.4 Eigencurves of newforms
We now recall Emerton’s eigenvariety construction (see sections 2.3 and 4 of [19]), which
we will apply to the space H1new(K(N, l), E)m, with the slight variation that we will use
Hecke operators at l as well. Applying the Jacquet functor (see [17]) to the locally ana-
lytic vectors in H1new(K(N, l), E)m we get an essentially admissible T (Qp) representation,
where T is a maximal torus of GL2. This corresponds to a coherent sheaf E on the rigid
analytic space T which classifies continuous characters of T (Qp). Let Tl be the abstract
Hecke algebra over O generated by Tq, Sq for primes q - plN and the operator Ul. The
algebra Tl acts on E , generating a coherent sheaf of algebras A on T . We have the relative
spectrum Spec(A ) of A over T , a rigid analytic space over E. We define Dnew to be the
reduced rigid analytic space Spec(A )red
We have the following corollary of Theorem 3.3.8:
Corollary 3.4.1. The space Dnew is equidimensional of dimension 2.
Proof. Fix a pro-p compact open subgroup of G, which is small enough so that KpK(N)
is neat. Now Theorem 3.3.8 implies that Hnew1 (K(N, l),O)m is a free O[[Kp]]-module, i.e.
isomorphic to O[[Kp]]r as an O[[Kp]]-module for some integer r. For ease of notation,
let V denote the G-representation H1new(K(N, l), E)m. By duality (and then inverting p),
V is isomorphic to C (Kp, E)r as a Kp-representation, where C (Kp, E) is the space of
continuous functions from Kp to E, viewed as a representation of Kp by letting Kp act
by right translation on functions. This shows that the space of locally analytic vectors Vlais isomorphic to C la(Kp, E)r as a Kp representation, where C la(Kp, E) is the space of
locally analytic functions from Kp to E. We let T0 denote the compact subgroup T ∩Kp
of T , and write M for the strong dual of the Jacquet module JB(Vla)′b, noting that M is the
space of global sections of the sheaf E over T . The rigid analytic variety T0 parameterises
the characters of T0, and there is a natural map T → T0 induced by restriction of characters.
The proof of Proposition 4.2.36 in [17] shows that, writing T0 as a union of admissi-
ble affinoid subdomains MaxSpec(An), there is an isomorphism of E{{z, z−1}}⊗EAn-
modules
Mn := M⊗C an(T0,E)An∼= E{{z, z−1}}⊗E[z](Wn⊗EAn),
whereWn is anE-Banach space and z is a certain element of T acting as a compact operator
onWn⊗EAn. Let Y be the subgroup of T generated by z, with corresponding rigid analytic
character variety Y . We can now describe how to apply the machinery of [3, 9] to get our
Chapter 3. Completed cohomology and level raising for p-adic modular forms 49
desired equidimensionality result. Let En be the pullback of E to T ×T0 An, let An be the
pullback of A to T ×T0 An, and let Zn ↪→ Y ×E An be the Fredholm variety cut out by
the characteristic power series of z acting on Wn⊗EAn. Since En is the sheaf associated
to Mn, there is a finite map Spec(An) → Zn. In the language of [3], Zn is the ‘spectral
variety’, and MaxSpec(An) is ‘weight space’.
An admissible cover C of Zn is constructed in section 4 of [3], so that for X ∈ C , with
image V in MaxSpec(An) under projection, the pullback of Spec(An) to Y is given by the
spectrum of a commutative algebra of endomorphisms (coming from the Hecke operators
and the action of T ) of a locally free, finite type O(V )-module. Now Lemme 6.2.10 of [9]
applies, in the same way as the proof of Proposition 6.4.2 in [9], to deduce that Spec(An) is
equidimensional of dimension 2, for all n, hence Spec(A ) is equidimensional of dimension
2.
It follows from Proposition 4.4.6 in [19] that Dnew is a product of an equidimension
1 space Dnew and weight space, by left exactness of the Jacquet functor and the fact
that H1new(K(N, l), E)m is stable under the action of C (T0, E) induced by the map la-
belled 4.2.5 in [19]. This fact holds since this twisting action is induced from a GL2(Af )-
equivariant action on the direct limit of completed cohomology spaces over all tame levels,
so it will preserve the kernel of i† (which was defined by means of degeneracy maps). Note
that we can also apply the same construction to H1(K(N, l), E)m to get an equidimension
1 space D.
We now return to the notation of the introduction. For each non-Eisenstein max-
imal ideal m of T(K(N, l)) let D(N, l)m be the equidimensional closed rigid subspace
of D(N, l)c whose points have residual Galois representation corresponding to the Hecke
character induced by m. Set D(N, l)newm = D(N, l)new
c ∩ D(N, l)m. There is a natural
isomorphism between D(N, l)m and D, since both spaces are reduced, equidimensional
and contain a Zariski dense set of points corresponding to the systems of Hecke eigenval-
ues arising from classical modular forms of tame level Γ1(N) ∩ Γ0(l) (see the proof of
Theorem 7.5.8 in [18]). Similarly, since Dnew is equidimensional, it is isomorphic to the
Zariski closure of the classical l-new points in D(N, l)m, so there is a closed embedding
Dnew ↪→ D(N, l)newm . We can now prove the theorem stated in the introduction.
Theorem 3.4.2. D(N, l)newc is equidimensional of dimension 1. It is equal to the Zariski
closure of the classical l-new points in D(N, l)c.
3.5 Addendum: a remark on the level at p 50
Proof. It is enough to show that the closed embedding Dnew ↪→ D(N, l)newm is an isomor-
phism (for each non-Eisenstein m). As in Proposition 4.7 of [10] the complement of the
image of this embedding is a union of irreducible components of dimension 0. Suppose
the complement Z is non-empty, containing a point x1. This point must lie over a point
x of D(N)m, since in D(N, l)m it lies in an irreducible component of dimension 1 which
only contains l-old classical points, since x1 ∈ Z . We can now compare the completed
cohomology side and the overconvergent modular form side, since we can characterise
‘old and new’ points like x1 using Hecke operators - they come from points x satisfying
Tl(x)2−(l+1)2Sl(x) = 0. On the cohomology side this corresponds to an eigenclass in the
kernel of i† · i (recall that this composition can be expressed in terms of Hecke operators),
so in particular x1 corresponds to an eigenclass in the kernel of i† - therefore it lies inDnew,
contradicting the assumption that x1 lies in the complement.
3.5 Addendum: a remark on the level at p
This chapter has concentrated on the local behaviour at a prime l 6= p of families of (finite
slope, overconvergent) p-adic modular forms. Of course, one is also interested in the be-
haviour at the prime p. A serious study of this question really requires the consideration of
p-adic representations of GL2(Qp) attached to p-adic modular forms, and the p-adic local
Langlands correspondence (see [20]). In this section we will limit ourselves to making a
few remarks about the ‘level’ at p of finite slope overconvergent p-adic modular forms.
Lemma 3.5.1. Let f ∈ Sk(Γ1(Npr), ε,C) be a newform, for some nebentypus ε, r ≥ 1,
p - N and suppose that f is a Hecke eigenform (for all the Hecke operators, including
Uq for primes q dividing Np). Suppose that Upf = αf , with α 6= 0. Let π denote the
cuspidal automorphic representation of GL2(AQ) generated by f . Then there are two
distinct possibilities for the local factor of π at p:
1. The local factor πp is a (ramified) principal series representation PS(µ1, µ2) (the
normalised induction of µ1× µ2) with µ1 unramified and µ2 conductor pr. In partic-
ular, the p-power part, εp, of ε has conductor pr.
2. The local factor πp is a special representation of conductor p. In particular, ε has
conductor prime to p and r = 1. The eigenvalue α satisfies α2 = εN(p)pk−2.
Chapter 3. Completed cohomology and level raising for p-adic modular forms 51
Proof. This follows from analysing the possibilities for the conductor c of εp. If c equals pr
then we are in the situation of case 1 (see the example in [8, pg. 119]). If c is smaller than
pr, then we are in the situation of case 2 by the third part of [28, Theorem 3].
For an integer N , coprime to p, let D(N)0 denote the (reduced) cuspidal tame level
Γ1(N) eigencurve. We define two subsets of the classical points of D(N)0. Firstly, denote
by Z p−PS the points whose associated automorphic representation have local factor at
p a principal series representation (unramified or ramified). Secondly, denote by Z p−Sp
the points whose associated automorphic representation have local factor at p a special
representation (which is then necessarily of conductor p, by lemma 3.5.1). Note that the
union Z p−PS ∪Z p−Sp is the set of classical points of D(N)0, by lemma 3.5.1.
Proposition 3.5.2. The set of points Z p−PS is Zariski dense in D(N)0.
Proof. We claim that any affinoid open neighbourhood U of a point z ∈ Z p−Sp contains a
point in Z p−PS. This claim establishes the proposition, since the classical points are Zariski
dense in D(N)0. To prove the claim, first shrink U so that the slope and nebentypus (with
conductor prime to p) of points in U are constant. For x ∈ U let α(x) denote the Upeigenvalue of the overconvergent modular form corresponding to the point x. By Lemma
3.5.1 we have vp(α(z)) = (k − 2)/2, where k is the weight of z. The neighbourhood U
contains points z′ with arbitrarily large classical weights k′, but vp(α(z′)) = (k − 2)/2 by
our assumption on constancy of slope. Pick z′ so that vp(α(z′)) < k′ − 1 (i.e. k′ > k/2)
and k′ 6= k. Then Coleman’s classicality criterion implies that z′ is a classical point, whilst
vp(α(z′)) 6= (k′ − 2)/2 so by Lemma 3.5.1 z′ must lie in Z p−PS.
Note that the proof of the above proposition in fact shows that given a sufficiently small
neighbourhood U of a point z ∈ Z p−Sp, all the classical points in U lie in Z p−PS. In other
words, Z p−Sp is closed in the p-adic topology. One might conjecture that it is also Zariski
closed. The following corollary of some results due to Chenevier [11] gives some evidence
towards this conjecture.
Proposition 3.5.3. Suppose N = 1 and p = 2, 3, 5 or 7. Then the set of p-special points
Z p−Sp is Zariski closed.
Proof. Let π be the map D(N)0 → W ×Gm sending a point x to the pair (κ(x), Up(x)−2),
where κ(x) is the weight of x and Up(x) is the Up-eigenvalue of x. The image of π is
a reduced curve X (note that this is not quite the spectral curve for Up since we took
3.5 Addendum: a remark on the level at p 52
Up(x)−2 as the second factor of the map). Let K be the set of integers k such that
dimC(Sk(Γ0(p),C)p−new) ≥ 2. Let Z p−SpK ⊂ Z p−Sp be the subset consisting of points
with weight in K . Its complement is a finite set, since the dimension of Sk(Γ0(p),C)p−new
grows with (even) k. It follows from the main theorem of [11] that π(Z p−SpK ) is precisely
the set of non-smooth points in X , whence we deduce that Z p−SpK is Zariski closed, and
therefore adding a finite number of additional points we see that Z p−Sp is also Zariski
closed.
53
Chapter 4
Completed cohomology of Shimura
curves and a p-adic Jacquet-Langlands
correspondence
4.1 Introduction 54
4.1 Introduction
In an important paper [38], Ribet proves the ‘ε conjecture’ of Serre, on optimising the level
of mod p Galois representations. A key result of Ribet’s paper establishes a relationship
between the arithmetic of Shimura curves and modular curves (Theorem 4.1 in [38]), which
provides a geometric realisation of an integral Jacquet-Langlands correspondence. More
recently, Rajaei [36] extended Ribet’s techniques to the case of Shimura curves over totally
real fields.
In this chapter we use Rajaei’s results together with Emerton’s completed cohomology
[19] to construct an integral p-adic Jacquet-Langlands map between completed cohomol-
ogy spaces for Shimura curves of different discriminant. Applying Emerton’s eigenvariety
construction to these spaces gives a Jacquet-Langlands map between eigenvarieties - we
note that our approach to proving this kind of ‘p-adic functoriality’ is rather different to
that followed in [10].
The techniques of this chapter also allow us to prove a p-adic analogue of Mazur’s
principle (as extended to the case of totally real fields by Jarvis, [25]), which should have
applications to questions of local-global compatibility at l 6= p in the p-adic setting - this
generalises Theorem 6.2.4 of [20] to totally real fields. Having proved a level lowering
result for completed cohomology, we can deduce a level lowering result for eigenvarieties.
For F = Q these level lowering results have already been obtained by Paulin [34], using
the results of [20] (in an indirect way, rather than by applying Emerton’s construction of
the eigencurve).
4.2 Preliminaries
4.2.1 Vanishing cycles
In this section we will follow the exposition of vanishing cycles given in the first chapter
of [36] - we give some details to fix ideas and notation. Let O be a characteristic zero
Henselian discrete valuation ring, residue field k of characteristic l, fraction field L. Let
X → S = Spec(O) be a proper, generically smooth curve with reduced special fibre, and
define Σ to be the set of singular points on X ⊗k. Moreover assume that a neighbourhood
of each point x ∈ Σ is etale locally isomorphic over S to Spec(O[t1, t2]/(t1t2 − ax)) with
ax a non-zero element of O with valuation ex = v(ax) > 0. Define morphisms i and j to
Chapter 4. Completed cohomology of Shimura curves 55
be the inclusions:
i : X ⊗ k →X , j : X ⊗ L→X .
For F a constructible torsion sheaf on X , with torsion prime to l, the Grothendieck (or
Leray) spectral sequence for the composition of the two functors Γ(X ,−) and j∗ gives an
identification
RΓ(X ⊗ L, j∗F ) = RΓ(X , R(j)∗j∗F ).
Applying the proper base change theorem we also have
RΓ(X , R(j)∗j∗F ) ' RΓ(X ⊗ k, i∗R(j)∗j
∗F ).
The natural adjunction id =⇒ R(j)∗j∗ gives a morphism i∗F → i∗R(j)∗j
∗F of com-
plexes on X ⊗ k. We denote by RΦ(F ) the mapping cone of this morphism (the complex
of vanishing cycles), and denote by RΨ(F ) the complex i∗R(j)∗j∗F (the complex of
nearby cycles). There is a distinguished triangle
i∗F → RΨ(F )→ RΦ(F )→,
whence a long exact sequence of cohomology
· · · → H i(X ⊗ k, i∗F )→ H i(X ⊗ k,RΨ(F ))→ H i(X ⊗ k,RΦ(F ))→ · · ·
As on page 36 of [36] there is a specialisation exact sequence, with ‘(1)’ denoting a Tate
twist of the Gal(L/L) action:
0 // H1(X ⊗ k, i∗F )(1) // H1(X ⊗ L, j∗F )(1)β //
⊕x∈Σ R
1Φ(F )x(1)
γ // H2(X ⊗ k, i∗F )(1)sp(1) // H2(X ⊗ L, j∗F )(1) // 0.
(4.1)
We can extend the above results to lisse etale Zp-sheaves (p 6= l), by taking inverse limits
of the exact sequences for the sheaves F/piF . Since our groups satisfy the Mittag-Leffler
condition we preserve exactness. In particular we obtain the specialisation exact sequence
(4.1) for F . From now on F will denote a Zp-sheaf, and we define
4.2 Preliminaries 56
X(F ) := ker(γ) = im(β) ⊂⊕x∈Σ
R1Φ(F )x(1),
so there is a short exact sequence
0 // H1(X ⊗ k, i∗F )(1) // H1(X ⊗ L, j∗F )(1) // X(F ) // 0.
As in section 1.3 of [36] there is also a cospecialisation sequence
0 //H0(X ⊗ k,RΨ(F ))
//H0(X ⊗ k,F )
γ′ //⊕
x∈Σ H1x(X ⊗ k,RΨ(F ))
β′ // H1(X ⊗ k,RΨ(F )) // H1(X ⊗ k,F ) // 0,
(4.2)
where we write X ⊗ k for the normalisation of X ⊗ k, and (in the above, and from now
on) we also denote by F the appropriate pullbacks of F and RΨ(F ) (e.g. i∗F ). The
normalisation map is denoted r : X ⊗ k →X ⊗ k. We can now define
X(F ) := im(β′),
and apply Corollary 1 of [36] to get the following:
Proposition 4.2.2. We have the following diagram made up of two short exact sequences:
0
��
X(F )
��
0 // H1(X ⊗ k,F ) //
��
H1(X ⊗ L,F ) // X(F )(−1) // 0
H1(X ⊗ k, r∗r∗F )
��0
Chapter 4. Completed cohomology of Shimura curves 57
Note that the map X(F )→ H1(X ⊗ k,F ) is the one induced by the fact that
X(F ) ⊂ H1(X ⊗ k,RΨ(F ))
is in the image of the injective map
H1(X ⊗ k,F )→ H1(X ⊗ k,RΨ(F )).
Finally, as in (1.13) of [36] there is an injective map
λ : X(F )→ X(F )
coming from the monodromy pairing.
4.3 Shimura curves and their bad reduction
4.3.1 Notation
Let F/Q be a totally real number field of degree d. We denote the infinite places of F by
τ1, ..., τd. We will be comparing the arithmetic of two quaternion algebras over F . The
first quaternion algebra will be denoted by D, and we assume that it is split at τ1 and at
all places in F dividing p, and non-split at τi for i 6= 1 together with a finite set S of
finite places. If d = 1 then we furthermore assume that D is split at some finite place
(so we avoid working with non-compact modular curves). Fix two finite places q1 and q2
of F , which do not divide p and where D is split, and denote by D′ a quaternion algebra
over F which is split at τ1 and non-split at τi for i 6= 1 together with each finite place
in S ∪ {q1, q2}. We fix maximal orders OD and OD′ of D and D′ respectively, and an
isomorphism D⊗F A(q1q2)F
∼= D′ ⊗F A(q1q2)F compatible with the choice of maximal orders
in D and D′.
We write q to denote one of the places q1, q2, and denote the completion Fq of F by L,
with ring of integersOq and residue field kq. Denote F ∩Oq byO(q). We have the absolute
Galois group Gq = Gal(L/L) with its inertia subgroup Iq.
Let G and G′ denote the reductive groups over Q arising from the unit groups of D and
D′ respectively. Note that G and G′ are both forms of ResF/Q(GL2/F ). For U a compact
open subgroup of G(Af ) and V a compact open subgroup of G′(Af ) we have complex
4.3 Shimura curves and their bad reduction 58
(disconnected) Shimura curves
M(U)(C) = G(Q)\G(Af )× (C− R)/U
M ′(U)(C) = G′(Q)\G′(Af )× (C− R)/V,
where G(Q) and G′(Q) act on C− R via the τ1 factor of G(R) and G′(R) respectively.
Both these curves have canonical models over F , which we denote by M(U) and
M ′(U).
4.3.2 Integral models and coefficient sheaves
Let U ⊂ G(Af ) be a compact open subgroup (we assume U is a product of local factors
over the places of F ), with Uq = U0(q) (the matrices in GL2(Oq) which have upper trian-
gular image in GL2(kq)), and U unramified at all the finite places v where D is non-split
(i.e. Uv = O×D,v). Let Σ(U) denote the set of finite places where U is ramified. In Theorem
8.9 of [25], Jarvis constructs an integral model for M(U) overOq which we will denote by
Mq(U). In fact this integral model only exists when U is sufficiently small (a criterion for
this is given in [25, Lemma 12.1]), but this will not cause us any difficulties (see the remark
at the end of this subsection). Section 10 of [25] shows that Mq(U) satisfies the conditions
imposed on X in section 4.2.1 above.
We also have integral models for places dividing the discriminant of our quaternion
algebra. Let V ⊂ G′(Af ) be a compact open subgroup with Vq = O×D′,q. We let M′q(V )
denote the integral model for M ′(V ) over Oq coming from Theorem 5.3 of [46] (this is
denoted C in [36]). This arises from the q-adic uniformisation of M ′(V ) by a formal Oq-
scheme, and gives integral models for varying V which are compatible under the natural
projections, and again satisfy the conditions necessary to apply section 4.2.1.
We now want to construct sheaves on Mq(U) and M′q(V ) corresponding to the possible
(cohomological) weights of Hilbert modular forms. Let k = (k1, . . . kd) be a d-tuple of
integers (indexed by the embeddings τi of F into R if the reader prefers), all≥ 2 and of the
same parity. The d-tuple of integers v is then characterised by having non-negative entries,
at least one of which is zero, with k+ 2v = (w, ..., w) a parallel vector. We will now define
p-adic lisse etale sheaves Fk on Mq(U), as in [7, 31, 39]. Denote by Z the centre of the
algebraic group G. Note that Z ∼= ResF/Q(Gm,F ). We let Zs denote the maximal subtorus
of Z that is split over R but which has no subtorus split over Q. It is straightforward to see
Chapter 4. Completed cohomology of Shimura curves 59
that Zs is the subtorus given by the kernel of the norm map
NF/Q : ResF/Q(Gm,F )→ Gm,Q.
In the notation of [31, Ch. III] the algebraic group Gc is the quotient of G by Zs. Take
F ′ ⊂ C a number field of finite degree, Galois over Q and containing F such that F ′ splits
both D and D′. Since F ′/Q is normal, F ′ contains all the Galois conjugates of F and we
can identify the embeddings {τi : F → C} with the embedding {τi : F → F ′}, via the
inclusion F ′ ⊂ C. Since D ⊗F,τi F ′ ∼= M2(F ′) for each i, we obtain representations ξi of
D× = G(Q) on the space Wi = F ′2. Let ν : G(Q)→ F× denote the reduced norm. Then
ξ(k) = ⊗di=1(τi ◦ ν)vi ⊗ Symki−2(ξi)
is a representation of G(Q) acting on Wk = ⊗di=1 Symki−2(Wi). In fact ξ(k) arises from an
algebraic representation of G on the Q-vector space underlying Wk (or rather the associ-
ated Q-vector space scheme). For z ∈ Z(Q) = F× the action of ξ(k)(z) is multiplication
by NF/Q(z)w−2, so ξ(k) factors through Gc. The representation ξ(k) gives rise to a repre-
sentation of G(Qp) on Wk ⊗Q Qp = Wk ⊗F ′ (F ′ ⊗Q Qp), so taking a prime p in F ′ above
p and projecting to the relevant factor of F ′ ⊗Q Qp we obtain an F ′p-linear representation
Wk,p of G(Qp).
Now we may define a Qp-sheaf on M(U)(C):
Fk,Qp,C = G(Q)\(G(Af )× (C− R)×Wk,p/U,
where G(Q) acts on Wk,p by the representation described above (via its embedding into
G(Qp)), and U acts only on the G(Af ) factor, by right multiplication. Note that this defi-
nition only works because ξ(k) factors through Gc.
Choose an OF ′p lattice W 0k,p in Wk,p. After projection to G(Qp), U acts on Wk,p, and
if U is sufficiently small it stabilises W 0k,p. Pick a normal open subgroup U1 ⊂ U acting
trivially on W 0k,p/p
n. We now define a Z/pnZ-sheaf on M(U)(C) by
Fk,Z/pnZ,C = (M(U1)(C)× (W 0k,p/p
n))/(U/U1).
4.3 Shimura curves and their bad reduction 60
Now on the integral model Mq(U) we can define an etale sheaf of Z/pnZ-modules by
Fk,Z/pnZ = (Mq(U1)× (W 0k,p/p
n))/(U/U1).
By section 2.1 of [7] this construction is independent of the choice of lattice W 0k,p. Taking
inverse limits of these Z/pnZ-sheaves we get a Zp- (in factOF ′p-) lisse etale sheaf on Mq(U)
which will be denoted by Fk.
The same construction applies to M′q(V ), and we also denote the resulting etale sheaves
on M′q(V ) by Fk. Note that we chose F ′ so that it split both D and D′ - this allows F ′p to
serve as a field of coefficients for both M′q(U) and M′q(V ).
From now on we work with a coefficient field E. This can be any finite extension of
F ′p. Denote the ring of integers in E by Op, and fix a uniformiser $. The sheaf Fk on the
curves Mq(U) and M′q(V ) will now denote the sheaves constructed above with coefficients
extended to Op.
Remark. Note that in the above discussion we assumed that the level subgroup U was
‘small enough’. In the applications below we will be taking direct limits of cohomology
spaces as the level at p, Up, varies over all compact open subgroups of G(Qp), so we can
always work with a sufficiently small level subgroup by passing far enough through the
direct limit.
4.3.3 Hecke algebras
Definition 4.3.4. The Hecke algebra Tk(U) is defined to be the Op-algebra of endomor-
phisms of H1(M(U)F ,Fk)E generated by Hecke operators Tv, Sv for primes v - pq1q2
such that U and D are both unramified at v. The Hecke algebra T′k(V ) is defined to be the
Op-algebra of endomorphisms of H1(M ′(V )F ,Fk)E generated by Hecke operators Tv, Sv
for primes v - p such that V and D′ are both unramified at v.
The Hecke algebras Tk(U) and T′k(V ) are semilocal, and their maximal ideals corre-
spond to mod p Galois representations arising from Hecke eigenforms occuring in
H1(M(U)F ,Fk)E andH1(M ′(V )F ,Fk)E respectively. Given a maximal ideal m of Tk(U)
we denote the Hecke algebra’s localisation at m by Tk(U)ρ, where ρ : Gal(F/F ) →GL2(Fp) is the mod p Galois representation attached to m, and more generally for any
Tk(U)-module M , denote its localisation at m by Mρ. We apply the same notational con-
vention to T′k(V ) and modules for this Hecke algebra.
Chapter 4. Completed cohomology of Shimura curves 61
The Hecke algebra Tk(U) acts faithfully on the Op-torsion free quotient
H1(M(U)F ,Fk)tf of H1(M(U)F ,Fk). Moreover, since the action of the Hecke operators
on H1(M(U)F ,Fk)tors is Eisenstein (see [15, Lemma 4]), if ρ as above is irreducible then
H1(M(U)F ,Fk)tfρ can be naturally viewed as anOp-direct summand of H1(M(U)F ,Fk),
and we denote it by H1(M(U)F ,Fk)ρ. The same consideration applies to any subquotient
M of H1(M(U)F ,Fk), and for the Hecke algebra T′k(V ). Note that if k = (2, ..., 2) then
H1(M(U)F ,Fk) = H1(M(U)F ,Op) is already Op-torsion free.
4.3.5 Mazur’s principle
In the next two subsections we summarise the results of [25] and [36]. First we fix a
(sufficiently small) compact open subgroup U ⊂ G(Af ) such that Uq = GL2(Oq). For
brevity we denote the compact open subgroup U ∩ U0(q) by U(q).
We apply the theory of Section 4.2.1 to the curve Mq(U(q)) and the Zp-sheaf given by
Fk, for some weight vector k, to obtainOp-modulesXq(U(q),Fk) and Xq(U(q),Fk). We
have short exact sequences
0→ H1(Mq(U(q))⊗ kq,Fk)→ H1(Mq(U(q))⊗ L,Fk)→ Xq(U(q),Fk)(−1)→ 0,
(4.3)
0→ Xq(U(q),Fk)→ H1(Mq(U(q))⊗ kq,Fk)→ H1(Mq(U(q))⊗ kq, r∗r∗Fk)→ 0.
(4.4)
The above short exact sequences are equivariant with respect to the Gq action, as well as
the action of Tk(U(q)) and the Hecke operator Uq. The sequence 4.4 gives rise to Mazur’s
principle (see [25] and Theorem 4.4.26 below).
4.3.6 Ribet-Rajaei exact sequence
We also apply the theory of Section 4.2.1 to the curve M′q(V ) and the sheaf Fk, where V ⊂G′(Af ) is a (sufficiently small) compact open subgroup which is unramified at q. We obtain
Op-modules Yq(V,Fk) and Yq(V,Fk),where Y and Y correspond toX and X respectively.
As in equation 3.2 of [36] these modules satisfy the (Gq and T′k(V ) equivariant) short exact
sequences
0 // H1(M′q(V )⊗ kq,Fk) // H1(M′q(V )⊗ L,Fk) // Yq(V,Fk)(−1) // 0,
4.3 Shimura curves and their bad reduction 62
0 // Yq(V,Fk) // H1(M′q(V )⊗ kq,Fk) // H1(M′q(V )⊗ kq, r∗r∗Fk) // 0.
It is remarked in section 3 of [36] that in fact H1(M′q(V )⊗ kq, r∗r∗Fk) = 0, since the
irreducible components of the special fibre of M′q(V ) are rational curves, so we have
Yq(V,Fk) ∼= H1(M′q(V )⊗ kq,Fk).
We can now relate our constructions for the two quaternion algebras D and D′. This
requires us to now distinguish between the two places q1 and q2. We make some further
assumptions on the levels U and V . First we assume that Uq1 = OD,q1 ∼= GL2(Oq1) and
Uq2 = OD,q2 ∼= GL2(Oq2), and that V is unramified at both q1 and q2. We furthermore
assume that the factors of U away from q1 and q2, U q1q2 , match with V q1q2 under the fixed
isomorphismD⊗FA(q1q2)F,f
∼= D′⊗FA(q1q2)F,f . We will use the notation U(q1), U(q2), U(q1q2)
to denote U ∩ U0(q1), U ∩ U0(q2), U ∩ U0(q1) ∩ U0(q2) respectively.
We fix a Galois representation ρ : Gal(F/F )→ GL2(Fp), satisfying two assumptions.
Firstly, we assume that ρ arises from a Hecke eigenform in H1(M ′(V )F ,Fk). Secondly,
we assume that ρ is irreducible. Equivalently, the corresponding maximal ideal m′ in T′k(V )
is not Eisenstein.
There is a surjection Tk(U(q1q2))→ T′k(V ) arising from the classical Jacquet–Langlands
correspondence (this identifies T′k(V ) as the q1q2-new quotient of Tk(U(q1q2))), so there
is a (non-Eisenstein) maximal ideal of Tk(U(q1q2)) corresponding to ρ (in fact this can be
established directly from the results contained on pg. 55 of [36], but for expository pur-
poses it is easier to rely on the existence of the Jacquet–Langlands correspondence). We
can therefore regard all T′k(V )ρ-modules as Tk(U(q1q2))ρ-modules. Now [36, Theorem 3]
implies the following:
Theorem 4.3.7. There are Tk(U(q1q2))ρ-equivariant short exact sequences
0 // Yq1(V,Fk)ρ // Xq2(U(q1q2),Fk)ρi† // Xq2(U(q2),Fk)
2ρ
// 0,
0 // Xq2(U(q2),Fk)2ρ
i // Xq2(U(q1q2),Fk)ρ // Yq1(V,Fk)ρ // 0.
In the theorem, the maps labelled i and i† are the natural ‘level raising’ map and its
adjoint, as in the previous chapters (they should be defined using double coset operators
here, since there is no direct moduli theoretic description of our Shimura curves allowing
a definition via degeneracy maps as in chapter 3). We think of this short exact sequence
Chapter 4. Completed cohomology of Shimura curves 63
as a geometric realisation of the Jacquet–Langlands correspondence between automorphic
forms for G′ and for G. The proof of the above theorem relies on relating the spaces
involved to the arithmetic of a third quaternion algebra D (which associated reductive al-
gebraic group G/Q) which is non-split at all the infinite places of F and at the places in
S ∪ {q2}. However, we will just focus on the groups G′ and G, and view G as playing an
auxiliary role.
4.4 Completed cohomology of Shimura curves
4.4.1 Completed cohomology
We now consider compact open subgroups Up of G(A(p)f ), with factor at v equal to O×Dv
at all places v of F where D is non-split. Then we have the following definitions, as in
[19, 20]:
Definition 4.4.2. We write HnD(Up,Fk/$
s) for the Op module
lim−→Up
Hn(M(UpUp)F ,Fk/$
s),
which has a smooth action of G(Qp) =∏
v|p GL2(Fv) and a continuous action of GF .
We then define a $-adically complete Op-module
Hn(Up,Fk) := lim←−s
Hn(Up,Fk/$s),
which has a continuous action of G(Qp) and GF .
Lemma 4.4.3. The Op[G(Qp)]-module HnD(Up,Fk) is a $-adically admissible G(Qp)-
representation over Op (in the sense of [21, Definition 2.4.7]).
Proof. This follows from [19, Theorem 2.2.11], taking care to remember the Op-integral
structure.
Write Hn(Up,Fk)E for the admissible continuous Banach E[G(Qp)]-representation
Hn(Up,Fk)⊗Op E.
Lemma 4.4.4. There is a canonical isomorphism
Hn(Up,Op) = lim←−s
Hn(Up,Op)/$sHn(Up,Op),
4.4 Completed cohomology of Shimura curves 64
compatible with GF and G(Qp) actions.
Proof. Since M(UpUp)F is proper of dimension 1, this follows from Corollary 2.2.27 of
[19]
Remark. Note that in Emerton’s notation the above lemma says that Hn(Up) = Hn(Up).
The following lemma allows us to restrict to considering completed cohomology with
trivial coefficients.
Lemma 4.4.5. There is a canonical G(Qp), GF equivariant isomorphism
Hn(Up,Fk) ∼= Hn(Up,Op)⊗Op W0k,p.
Proof. This follows from Theorem 2.2.17 of [19].
Proposition 4.4.6. Let F× be the closure of F× in Af,×F . There is an isomorphism between
H0(Up,Op) and the space of continuous functions
C (F×\Af,×F / det(Up),Op),
with GL2(Qp) action on the right hand side induced by g acting on Af,×F by multiplication
by det(g).
Proof. This follows from the fact that the product of the sign map on (C − R) and the
reduced norm map on G(Af ) induces a map
G(Q)\G(Af )× (C− R)/U → F×\A×F,f × {±}/ det(U) = F×\A×F,f/ det(U)
with connected fibres.
Proposition 4.4.7. The space H2(Up,Op) is equal to 0.
Proof. Exactly as for Proposition 4.3.6 of [19]
If v is a prime in F such that Up and D are both unramified at v then there are Hecke
operators Tv, Sv defined as usual on the spaces H1(Up,Op/$s) (i.e. by double coset oper-
ators, as in [36]). By continuity this extends to an action of these Hecke operators on the
space H1D(Up,Op), so we can make the following definition:
Chapter 4. Completed cohomology of Shimura curves 65
Definition 4.4.8. The Hecke algebra T(Up) is defined to be lim←−Up T(2,...,2)(UpUp). We en-
dow it with the projective limit topology given by taking the$-adic topology on each finitely
generated Op-algebra T(2,...,2)(UpUp).
The topological Op-algebra T(Up) acts faithfully on H1D(Up,Op), and is topologically
generated by Hecke operators Tv, Sv for primes v - p such that Up and D are both un-
ramified at v. Note that the isomorphisms of lemmas 4.4.4 and 4.4.5 are equivariant with
respect to the Hecke action, so we will often just work with the completed cohomology
spaces defined with trivial coefficients.
Definition 4.4.9. An ideal I of T(Up) is Eisenstein if the map λ : T (Up) → T (Up)/I
satisfies λ(Tq) = ε1(q) + ε2(q) and λ(qSq) = ε1(q)ε2(q) for all q /∈ {p} ∪ Σ, for some
characters ε1, ε2 : Z×pΣ → T (Up)/I .
The Hecke algebra T(Up) is semilocal and Noetherian, with maximal ideals corre-
sponding to mod p Galois representations arising from Hecke eigenforms in (an extension
of scalars of) H1D(Up,Op/$). Given a maximal ideal m of T(Up) we denote the Hecke
algebra’s localisation at m by T(Up)ρ, where ρ : Gal(F/F ) → GL2(Fp) is the mod p
Galois representation attached to m, and more generally for any T(Up)-module M , denote
its localisation at m by Mρ.
We now fix such a ρ, and assume that it is irreducible. There is a deformation ρmρ,Up of
ρ to T(Up)ρ (taking a projective limit, over compact open subgroups Up of G(Qp), of the
deformations to T(2,...,2)(UpUp)ρ). For a closed point P ∈ Spec(T(Up)ρ[1/p]) we denote
by ρ(P) the attached Galois representation, defined over the characteristic 0 field k(P).
Definition 4.4.10. Given a T(Up)ρ moduleM and a closed point P ∈ Spec(T(Up)ρ[1/p]),
we denote by M [P] the k(P)-vector space {m ∈ M ⊗Op k(P) : Tm = 0∀T ∈ P}.Suppose we have a homomorphism λ : T(Up)ρ → E ′, where E ′/E is a finite field ex-
tension. We say that the system of Hecke eigenvalues λ occurs in a T(Up)ρ-module M if
M ⊗Op E′ contains a non-zero element where T(Up) acts via the character λ. In other
words, λ occurs in M if and only if M [ker(λ)] 6= 0. We say that λ is Eisenstein whenever
kerλ is.
We have the following result, generalising [20, Proposition 5.5.3]. Denote by X the
Op-module HomT(Up)ρ[GF ](ρmρ,Up , H
1D(Up,Op)ρ).
4.4 Completed cohomology of Shimura curves 66
Proposition 4.4.11. The natural evaluation map
ev : ρmρ,Up ⊗T(Up)ρ X → H1D(Up,Op)ρ
is an isomorphism.
Proof. First let P ∈ Spec(T(Up)ρ[1/p]) be the prime ideal of the Hecke algebra corre-
sponding to a classical point coming from a Hecke eigenform in H1(M(UpUp),Op)ρ ⊗Op
k(P). Then the image of the map
ρ(P)⊗k(P) X[P]→ H1D(Up,Op)ρ ⊗Op k(P)
induced by ev contains H1(M(UpUp)ρ,Op)[P]. From this, it is straightforward to to see
that the image of the map
evE : E ⊗Op ρmρ,Up ⊗T(Up)ρ X → H1
D(Up,Op)ρ,E
is dense, since it will contain the dense subspace H1D(Up,Op)ρ,E . Now X is a $-adically
admissible G(Qp)-representation over T(Up)ρ, hence by [20, Proposition 3.1.3] the image
of evE is closed, therefore evE is surjective. We now note that the evaluation map
ρ⊗k(p) (HomGF (ρ,H1D(Up,Op/$)ρ)→ H1
D(Up,Op/$)ρ
is injective, by the irreducibility of ρ. We conclude the proof of the proposition by applying
the following lemma (see [20, Lemma 3.1.6]).
Lemma 4.4.12. Let π1 and π2 be two $-adically admissible G(Qp) representations over a
complete local NoetherianOp-algebra A (with maximal ideal m), which are torsion free as
Op-modules. Suppose f : π1 → π2 is a continuous A[G(Qp)]-linear morphism, satisfying
1. The induced map π1 ⊗Op E → π2 ⊗Op E is a surjection
2. The induced map π1/$[m]→ π2/$[m] is an injection.
Then f is an isomorphism.
Proof. First we show that f is an injection. Denote the kernel of f by K. It is a $-adically
admissible G(Qp) representation over A, so in particular K/$ =⋃i≥1K/$[mi]. Our
Chapter 4. Completed cohomology of Shimura curves 67
second assumption on f then implies that K/$ = 0, which implies K = 0, since K is
$−adically separated. We now know that f is an injection, and out first assumption on
f implies that it has Op-torsion cokernel C. Since π1 is Op-torsion free there is an exact
sequence
0→ C[$]→ π1/$ → π2/$.
Since π1/$ =⋃i≥1 π1/$[mi] we have C[$] =
⋃i≥1C[$][mi], but the second assumption
on f implies that C[$][m] = 0. Therefore C[$] = 0, and since C is Op-torsion we have
C = 0. Therefore f is an isomorphism.
4.4.13 Completed vanishing cycles
We now assume that our tame level Up is unramified at the prime q, and let Up(q) denote
the tame level Up ∩ U0(q). Fix a non-Eisenstein maximal ideal m of T(Up(q)). We are
going to ‘$-adically complete’ the constructions of subsection 4.3.5.
Definition 4.4.14. For F equal to either Fk or r∗r∗Fk we defineOp-modules with smooth
G(Qp) actions
H1red(Up(q),F ) = lim−→
Up
H1(Mq(UpUp(q))⊗ kq,F ),
Xq(Up(q),Fk) = lim−→
Up
Xq(UpUp(q),Fk),
and
Xq(Up(q),Fk) = lim−→
Up
Xq(UpUp(q),Fk).
Proposition 4.4.15. Equation 4.3 induces a short exact sequence, equivariant with respect
to Hecke, G(Qp) and Gq actions:
0 // H1red(Up(q),Fk) // H1(Up(q),Fk) // Xq(U
p(q),Fk)(−1) // 0.
Similarly, Equation 4.4 induces
0 // Xq(Up(q),Fk) // H1
red(Up(q),Fk) // H1red(Up(q), r∗r
∗Fk) // 0.
Proof. This just follows from taking the direct limit of the relevant exact sequences.
4.4 Completed cohomology of Shimura curves 68
Definition 4.4.16. We define Op-modules with continuous G(Qp) actions (again F equals
either Fk or r∗r∗Fk)
H1red(Up(q),F ) = lim←−
s
H1red(Up(q),F )/$sH1
red(Up(q),F ),
Xq(Up(q),Fk) = lim←−
s
Xq(Up(q),Fk)/$
sXq(Up(q),Fk)
and ˜Xq(Up(q),Fk) = lim←−
s
Xq(Up(q),Fk)/$
sXq(Up(q),Fk).
Now $-adically completing the short exact sequences of Proposition 4.4.15, we obtain
Proposition 4.4.17. We have short exact sequences of $-adically admissible
G(Qp)-representations overOp, equivariant with respect to Hecke, G(Qp) and Gq actions:
0 // H1red(Up(q),Fk)m
// H1D(Up(q),Fk)m
// Xq(Up(q),Fk)m(−1) // 0,
0 // ˜Xq(Up(q),Fk)m
// H1red(Up(q),Fk)m
// H1red(Up(q), r∗r
∗Fk)m// 0.
Proof. The existence of a short exact sequence of Op-modules follows from the fact that
Xq(UpUp(q),Fk)m and H1
red(Up(q), r∗r∗Fk)m is always Op-torsion free. The module
Xq(UpUp(q),Fk) is in fact torsion free even without localising at m (see the remark follow-
ing [36, Corollary 1]), whilst for H1red(Up(q), r∗r
∗Fk)m this fact follows as in [15, Lemma
4]. The fact that we obtain a short exact sequence of $-adically admissible Op[G(Qp)]-
modules follows from Proposition 2.4.4 of [21].
It is now straightforward to deduce the analogue of Lemma 4.4.5:
Lemma 4.4.18. There are canonical G(Qp), Gq and Hecke equivariant isomorphisms
H1red(Up(q),Fk)m ∼= H1
red(Up(q),Op)m ⊗Op W0k,p,
˜Xq(Up(q),Fk)m ∼= ˜Xq(U
p(q),Op)m ⊗Op W0k,p,
Xq(Up(q),Fk)m ∼= Xq(U
p(q),Op)m ⊗Op W0k,p.
Chapter 4. Completed cohomology of Shimura curves 69
4.4.19 Mazur’s principle
We can now proceed as in Theorem 6.2.4 of [20], following Jarvis’s approach to Mazur’s
principle over totally real fields [25]. First we fix an absolutely irreducible mod p Ga-
lois representation ρ, coming from a maximal ideal of T(Up(q)). We have the following
proposition
Proposition 4.4.20. The injection
H1red(Up(q),Op) ↪→ H1
D(Up(q),Op)
induces an isomorphism
H1red(Up(q),Op)ρ ∼= (H1
D(Up(q),Op)ρ)Iq .
Proof. This follows from Proposition 4 and (2.2) in [36].
There is a Hecke operator Uq acting on H1(Up(q),Op), which induces an action on the
spaces Xq(Up(q),Op) and Xq(U
p(q),Op). This extends by continuity to give an action of
Uq on ˜Xq(Up(q),Op) and Xq(U
p(q),Op).
Lemma 4.4.21. The Frobenius element of Gal(kq/kq) acts on ˜Xq(Up(q),Op) by (Nq)Uq
and on Xq(Up(q),Op) by Uq.
Proof. Section 6 of [7] tells us that Frobenius acts as required on the dense subspaces
Xq(Up(q),Op) and Xq(U
p(q),Op), so by continuity we are done.
Lemma 4.4.22. There is a natural isomorphism H1red(Up(q), r∗r
∗Op) ∼= H1D(Up,Op)
⊕2.
Proof. This follows from Lemma 16.1 in [25].
Lemma 4.4.23. The monodromy pairing map λ : Xq(Up(q),Op) → Xq(U
p(q),Op) in-
duces a Gq- and T(Up(q))-equivariant isomorphism Xq(Up(q),Op)ρ ∼= ˜Xq(U
p(q),Op)ρ.
Proof. This follows from Proposition 5 in [36].
Remark. Note that the isomorphism in the above lemma is not Uq equivariant.
Before giving an analogue of Mazur’s principle, we will make explicit the connection
between the modules Xq(Up(q),Op)ρ and a space of newforms. There are natural level rais-
ing maps i : H1D(Up,Op)
⊕2ρ → H1
D(Up(q),Op)ρ, and their adjoints i† : H1D(Up(q),Op)ρ →
H1D(Up,Op)
⊕2ρ , defined as in the previous chapter by taking the limit of maps at finite level.
4.4 Completed cohomology of Shimura curves 70
Definition 4.4.24. The space of q-newforms
H1D(Up(q),Op)
q−newρ
is defined to be the kernel of the map
i† : H1D(Up(q),Op)ρ → H1
D(Up,Op)⊕2ρ .
Proposition 4.4.25. A system of Hecke eigenvalues λ : T(Up(q)) → E ′ (E ′/E finite)
occurs in H1D(Up(q),Op)
q−newρ if and only if it occurs in Xq(U
p(q),Op)ρ.
Proof. Proposition 4.4.17 and Lemma 4.4.22, together with Definition 4.4.24 imply that we
have the following commutative diagram of $-adically admissible G(Qp)-representations
over Op:
0 // ˜Xq(Up(q),Op)ρ
//
α
��
H1red(Up(q),Op)ρ //
β
��
H1D(Up,Op)
⊕2ρ
//
∼=��
0
0 // H1D(Up(q),Op)
q−newρ
// H1D(Up(q),Op)ρ // H1
D(Up,Op)⊕2ρ
// 0,
where the two rows are exact, the maps α and β are injections and the final vertical map
is the identity. Moreover coker(β) = Xq(Up(q),Op)ρ(−1) ∼= ˜Xq(U
p(q),Op)ρ(−1), where
the second isomorphism comes from the monodromy pairing (see Lemma 4.4.23). Apply-
ing the snake lemma we see that coker(α) = coker(β), so (applying Lemma 4.4.23 once
more to the source of the map α) we have a short exact sequence
0 // Xq(Up(q),Op)ρ
α // H1D(Up(q),Op)
q−newρ
// Xq(Up(q),Op)ρ(−1) // 0.
From this short exact sequence it is easy to deduce that the systems of Hecke eigenvalues
occurring in H1D(Up(q),Op)
q−newρ and Xq(U
p(q),Op)ρ are the same.
Theorem 4.4.26. Let ρ : Gal(F/F ) → GL2(Qp) be a Galois representation, with irre-
ducible reduction ρ : Gal(F/F ) → GL2(Fp). Suppose the following two conditions are
verified:
1. ρ is unramified at the prime q
2. ρ(Frobq) is not a scalar
Chapter 4. Completed cohomology of Shimura curves 71
3. There is a system of Hecke eigenvalues λ : T(Up)ρ → E ′ attached to ρ (i.e. ρ ∼=ρ(ker(λ))), with λ occurring in H1
D(Up(q),Op).
Then λ occurs in H1D(Up,Op).
Proof. We take ρ as in the statement of the theorem. The third assumption, combined with
Proposition 4.4.11 implies that there is a prime ideal P = ker(λ) of T(Up(q)), and an
embedding
ρ ↪→ H1D(Up(q),Op)[P].
Our aim is to show that H1D(Up,Op)[P] 6= 0.
Let T(Up(q))∗ denote the finite ring extension of T(Up(q)) obtained by adjoining the
operator Uq. Since ρ is irreducible there is a prime P∗ of T(Up(q))∗ such that there is an
embedding
ρ ↪→ H1D(Up(q),Op)[P
∗].
Recall that by Propositions 4.4.17 and 4.4.20 and Lemma 4.4.22 there is a Gq and
T(Up(q))-equivariant short exact sequence
0 // ˜Xq(Up(q),Op)ρ
// H1D(Up(q),Op)
Iqρ
// H1D(Up,Op)
⊕2ρ
// 0.
Suppose for a contradiction that H1D(Up,Op)ρ[P] = 0
Let V denote theGq-representation obtained from ρ by restriction. Since ρ is unramified
at q, there is an embedding V ↪→ H1D(Up(q),Op)
Iqρ [P∗]. The above short exact sequence
then implies that the embedding
˜Xq(Up(q),Op)ρ[P] ↪→ H1
D(Up(q),Op)Iqρ [P]
is an isomorphism, and hence the embedding
˜Xq(Up(q),Op)ρ[P
∗] ↪→ H1D(Up(q),Op)
Iqρ [P∗]
is also an isomorphism.
This implies that there is an embedding V ↪→ ˜Xq(Up(q),Op)ρ[P∗], so by Lemma
4.4.21 ρ(Frobq) is a scalar (Nq)α where α is the Uq eigenvalue coming from P∗, contra-
dicting our second assumption.
4.4 Completed cohomology of Shimura curves 72
4.4.27 Mazur’s principle and the Jacquet functor
After taking locally analytic vectors and applying Emerton’s locally analytic Jacquet func-
tor (see [17]) we can prove a variant of Theorem 4.4.26. We let B denote the Borel sub-
group of G(Qp) consisting of upper triangular matrices, and let T denote the maximal
torus contained in B. We then have a locally analytic Jacquet functor JB as defined in [17],
which can be applied to the space of locally analytic vectors (V )an of an admissible contin-
uous Banach E[G(Qp)]-representation V . The following lemma follows from Proposition
4.4.11, but we give a more elementary separate proof.
Lemma 4.4.28. Let λ : T(Up(q)) → E be a system of Hecke eigenvalues such that λ oc-
curs in JB(H1D(Up(q), E)an). Suppose the attached Galois representation ρ is absolutely
irreducible.Then there is a non-zero Gal(F/F )-equivariant map (necessarily an embed-
ding, since ρ is irreducible)
ρ ↪→ JB(H1D(Up(q), E)an).
Proof. To abbreviate notation we let M denote JB(H1D(Up(q), E)an). Since M is an es-
sentially admissible T representation, the system of Hecke eigenvalues λ occurs in the
χ-isotypic subspace Mχ for some continuous character χ of T . So (again by essential ad-
missibility) Mχ,λ is a non-zero finite dimensional Gal(F/F )-representation over E, and
therefore by the Eichler-Shimura relations (as in section 10.3 of [6]) and the irreducibility
of ρ the desired embedding exists.
Theorem 4.4.29. Suppose that the system of Hecke eigenvalues λ : T(Up(q)) → E
occurs in JB(H1D(Up(q), E)an), that the attached Galois representation ρ : Gal(F/F ) →
GL2(E) is unramified at q, with ρ(Frobq) not a scalar, and that the reduction
ρ : Gal(F/F )→ GL2(Fp) is irreducible.
Then λ occurs in JB(H1D(Up, E)an).
Proof. We take ρ as in the statement of the theorem. Let m be the maximal ideal of
T(Up(q)) attached to ρ, and let P be the prime ideal of T(Up(q)) attached to ρ, so ap-
plying Lemma 4.4.28 gives an embedding
ρ ↪→ JB(H1D(Up(q), E)anρ )[P].
Our aim is to show that JB(H1D(Up, E)anρ )[P] 6= 0. Let T(Up(q))∗ denote the finite ring
Chapter 4. Completed cohomology of Shimura curves 73
extension of T(Up(q)) obtained by adjoining the operator Uq. Since ρ is irreducible there
is a prime P∗ of T(Up(q))∗ such that there is an embedding
ρ ↪→ JB(H1D(Up(q), E)anρ )[P∗].
By Propositions 4.4.17 and 4.4.20 and Lemma 4.4.22 there is a short exact sequence
0 // ˜Xq(Up(q), E)ρ
// H1D(Up(q), E)
Iqρ
// H1D(Up, E)2
ρ// 0.
Taking locally analytic vectors (which is exact) and applying the Jacquet functor (which is
left exact), we get an exact sequence
0 // JB( ˜Xq(Up(q), E)anρ ) // JB((H1
D(Up(q), E)Iqρ )an) // JB(H1
D(Up, E)anρ )2.
Note that JB((H1D(Up(q), E)
Iqρ )an) is equal to JB(H1
D(Up(q), E)anρ )Iq , since under the
isomorphism of Proposition 4.4.11 they both correspond to (ρmρ,Up(q))Iq⊗T(Up(q))ρ JB(Xan
E ).
Let V denote the Gq-representation obtained from ρ by restriction. Since ρ is unramified at
q, there is an embedding V ↪→ JB((H1D(Up(q), E)
Iqρ )an)[P∗]. Suppose for a contradiction
that JB(H1D(Up, E)anρ )[P] = 0. The above exact sequence then implies that the embedding
JB( ˜Xq(Up(q), E)anρ )[P] ↪→ JB((H1
D(Up(q), E)Iqρ )an)[P]
is an isomorphism, and hence the embedding
JB( ˜Xq(Up(q), E)anρ )[P∗] ↪→ JB((H1
D(Up(q), E)Iqρ )an)[P∗]
is also an isomorphism.
This implies that there is an embedding V ↪→ JB( ˜Xq(Up(q), E)anρ )[P∗], so by Lemma
4.4.22 ρ(Frobq) is a scalar (Nq)α corresponding to the (Nq)Uq eigenvalue in P∗. We then
obtain a contradiction exactly as in the proof of Theorem 4.4.26.
4.4.30 A p-adic Jacquet–Langlands correspondence
We now wish to relate the arithmetic of p-adic automorphic forms for G and G′. The
definitions and results of subection 4.4.1 apply to G′. We denote the associated completed
4.4 Completed cohomology of Shimura curves 74
cohomology spaces of tame level V p by
H1D′(V
p,Fk),
and its Hecke algebra by T′(V p). The integral models M′q and the vanishing cycle formal-
ism also allows us to define some more topological Op-modules:
Definition 4.4.31. We define Op-modules with smooth G(Qp) actions
Yq(Vp,Fk) = lim−→
Vp
Yq(VpVp,Fk),
and
Yq(Vp,Fk) = lim−→
Vp
Yq(VpVp,Fk).
We define Op-modules with continuous G(Qp) actions
Yq(Vp,Fk) = lim←−
s
Y (V p,Fk)/$sY (V p,Fk)
and ˜Y q(Vp,Fk) = lim←−
s
Yq(Vp,Fk)/$
sYq(Vp,Fk).
Fixing a non-Eisenstein maximal ideal m′ of T′(V p), then in exactly the same way as
we obtained Proposition 4.4.17 we get:
Proposition 4.4.32. We have a short exact sequence of $-adically admissible G(Qp)-
representations over Op, equivariant with respect to T′(V p)m′ , G(Qp) and Gq actions:
0 // ˜Y q(Vp,Fk)m′
// H1D′(V
p,Fk)m′// Yq(V
p,Fk)m′(−1) // 0.
Fix Up ⊂ G(A(p)f ) such that Up is unramified at the places q1, q2, and matches with
V p at all other places. We now fix an irreducible mod p Galois representation ρ, arising
from a (necessarily non-Eisenstein) maximal ideal m′ of T′(V p). There is a correspond-
ing maximal ideal m′0 of T′(2,...,2)(VpVp) for Vp a small enough compact open subgroup
of G′(Qp), and this pulls back by the map T(2,...,2)(Up(q1q2)Up) → T′(2,...,2)(V
pVp) to a
maximal ideal m0 of T(2,...,2)(Up(q1q2)Up) for Up the compact open subgroup of G(Qp)
which is identified with Vp under the isomorphism G(Qp) ∼= G′(Qp). Hence there is a map
T(Up(q1q2))ρ → T′(V p)ρ.
Chapter 4. Completed cohomology of Shimura curves 75
We can give a refinement of Proposition 4.4.11 in this situation. Firstly we note that
the Galois representation ρmρ,V p : GF → GL2(T′(V p)ρ ⊗Op E) has an unramified sub-Gq
representation V0,E (which is a free T′(V p)ρ ⊗Op E-module of rank one), with unrami-
fied quotient V1,E∼= V0,E(−1) (also a free T′(V p)ρ ⊗Op E-module of rank one). This
follows from the fact that each representation ρ : GF → GL2(T′(2,...,2)(VpVp)ρ ⊗Op E)
has a unique unramified subrepresentation of the right kind. Denote by Y the Op-module
HomT′(V p)ρ[GF ](ρmρ,V p , H
1D′(V
p,Op)ρ). By proposition 4.4.11 we have an isomorphism
evE : E ⊗Op ρmρ,V p ⊗T′(V p)ρ Y
∼= H1D′(V
p,Op)ρ.
Proposition 4.4.33. The isomorphism evE identifies the image of the embedding
˜Y q(Vp,Op)ρ,E → H1
D′(Vp,Op)ρ,E
with V0,E ⊗T′(V p)ρ Y . It follows that the quotient Yq(V p,Op)ρ,E(−1) is identified with
V1,E ⊗T′(V p)ρ Y .
Proof. We proceed as in the proof of proposition 4.4.11. First let P ∈ Spec(T′(V p)ρ ⊗Op
E) be the prime ideal of the Hecke algebra corresponding to a classical point coming from
a Hecke eigenform in H1(M ′(V pVp),Op)ρ ⊗Op k(P). Taking Vp invariants, extending
scalars to k(P) and taking the P torsion parts, the isomorphism evE induces an isomor-
phism
ρ(P)⊗k(P)Homk(P)[GF ](ρ(P), H1(M ′(V pVp),Op)ρ,E[P]) ∼= H1(M ′(V pVp),Op)ρ,E[P],
under which Yq(V pVp,Op)ρ,E[P] is identified with
V0,E(P)⊗k(P) Homk(P)[GF ](ρ(P), H1(M ′(V pVp),Op)ρ,E[P]).
A density argument now completes the proof.
Passing to the direct limit over compact open subgroups Up of G(Qp) and $-adically
completing the short exact sequences of Theorem 4.3.7, we obtain
Theorem 4.4.34. We have the following short exact sequences of $-adically admissible
G(Qp)-representations over Op, equivariant with respect to T(Up(q1q2))ρ and G(Qp) ∼=
4.4 Completed cohomology of Shimura curves 76
G′(Qp) actions:
0 // Yq1(Vp,Op)ρ // Xq2(U
p(q1q2),Op)ρi† // Xq2(U
p(q2),Op)2ρ
// 0,
0 // ˜Xq2(Up(q2),Op)
2ρ
// ˜Xq2(Up(q1q2),Op)ρ
// ˜Y q1(Vp,Op)ρ
// 0.
Proof. The proof is as for Proposition 4.4.17. We just need to check that the short exact
sequences of Theorem 4.3.7 are compatible as Up and Vp vary, but this is clear from the
proof of [36, Theorem 3], since the descriptions of the dual graph of the special fibres of
Mq2(Up(q1q2)Up) and M′q1(V
pVp) are compatible as Up and Vp vary.
We regard Theorem 4.4.34 (especially the first exact sequence therein) as a geomet-
ric realisation of a p-adic Jacquet–Langlands correspondence. The following propositions
explain why this is reasonable. We let Xq2(Up(q1q2),Op)
q1−newρ denote the kernel of the
map
i† : Xq2(Up(q1q2),Op)ρ → Xq2(U
p(q2),Op)⊕2ρ .
Proposition 4.4.35. A system of Hecke eigenvalues λ : T(Up(q1q2)) → E ′ occurs in
H1D(Up(q1q2),Op)
q1q2−newρ if and only if it occurs in Xq2(U
p(q1q2),Op)q1−newρ .
Proof. We have the following commutative diagram:
0 // Xq2 (Up(q1q2),Op)q1−newρ
//
α
��
Xq2 (Up(q1q2),Op)ρ //
β
��
Xq2 (Up(q2),Op)⊕2ρ
//
γ
��
0
0 // H1D(Up(q1q2),Op)q1q2−new
ρ// H1
D(Up(q1q2),Op)q2−newρ
// (H1D(Up(q2),Op)q2−new
ρ )⊕2 ,
where the two rows are exact, and the maps α, β and γ are injections. Note that we are
applying Lemma 4.4.23. By the snake lemma and the proof of Proposition 4.4.25 we have
an exact sequence
0 // coker(α) // Xq2(Up(q1q2),Op)ρ(−1) // Xq2(U
p(q2),Op)⊕2ρ (−1),
whence a short exact sequence
0 −→ Xq2 (Up(q1q2),Op)q1−newρ
α−→ H1D(Up(q1q2),Op)q1q2−new
ρ −→ Xq2 (Up(q1q2),Op)q1−newρ (−1) −→ 0.
From this last short exact sequence it is easy to deduce that the systems of Hecke eigen-
values occurring in H1D(Up(q1q2),Op)
q1q2−newρ and Xq2(U
p(q1q2),Op)q1−newρ are the same.
Chapter 4. Completed cohomology of Shimura curves 77
Proposition 4.4.36. A system of Hecke eigenvalues λ : T′(V p)→ E ′ occurs in H1D′(V
p,Op)ρ
if and only if it occurs in Yq1(Vp,Op)ρ.
Proof. This follows from proposition 4.4.33.
Combined with Theorem 4.4.34 this has as a consequence
Corollary 4.4.37. The first exact sequence of Theorem 4.4.34 induces a bijection between
the systems of Hecke eigenvalues (for T′(V p)) appearing in H1D′(V
p,Op)ρ and the systems
of Hecke eigenvalues (for T(Up(q1q2))) appearing in H1D(Up(q1q2),Op)
q1q2−newρ .
4.5 Eigenvarieties and an overconvergent Jacquet–Langlands corre-
spondence
In this section we will apply Emerton’s construction of eigenvarieties to deduce an over-
convergent Jacquet–Langlands correspondence from Theorem 4.4.34.
4.5.1 Locally algebraic vectors
Fix a maximal ideal m of T(Up(q)) which is not Eisenstein. Recall that given an admissible
G(Qp)-representation V over E, Schneider and Teitelbaum [41] define an exact functor by
passing to the locally Qp-analytic vectors V an, and the G(Qp)-action on V an differentiates
to give an action of the Lie algebra g of G(Qp).
Proposition 4.5.2. There are natural isomorphisms (for F = Fk or r∗r∗Fk)
1. H0(g, H1D(Up(q),Fk)
anm,E) ∼= H1
D(Up(q),Fk)m,E,
2. H1(g, H1D(Up(q),Fk)
anm,E) ∼= 0,
3. H0(g, H1red(Up(q),F )anm,E) ∼= H1
red(Up(q),F )m,E,
4. H1(g, H1red(Up(q),F )anm,E) ∼= 0,
5. H0(g, Xq(Up(q),Fk)
anm,E) ∼= Xq(U
p(q),Fk)m,E,
6. H0(g, ˜Xq(Up(q),Fk)
anm,E) ∼= Xq(U
p(q),Fk)m,E,
4.5 Eigenvarieties and an overconvergent Jacquet–Langlands correspondence 78
Proof. The first and second natural isomorphisms comes from the spectral sequence (for
completed etale cohomology) discussed in Proposition 2.4.1 of [19], since localising at m
kills H iD(Up(q),Fk) and H i
D(Up(q),Fk) for i = 0, 2.
The same argument works for the third and fourth isomorphisms, since as noted in
the proof of [19, Proposition 2.4.1] the spectral sequence of [19, Corollary 2.2.18] can be
obtained using the Hochschild-Serre spectral sequence for etale cohomology, which applies
equally well to the etale cohomology of the special fibre of Mq(UpUp(q)).
We now turn to the fifth isomorphism. By the first short exact sequence of Proposition
4.4.17 and the exactness of localising at m, inverting $ and then passing to locally analytic
vectors, there is a short exact sequence
0 // H1red(Up(q),Fk)
anm,E
// H1D(Up(q),Fk)
anm,E
// Xq(Up(q),Fk)(−1)anm,E
// 0.
Taking the long exact sequence of g-cohomology then gives an exact sequence
0 // H0(g, H1red(Up(q),Fk)
anm,E) // H0(g, H1
D(Up(q),Fk)anm,E)
// H0(g, Xq(Up(q),Fk)(−1)anm,E) // H1(g, H1
red(Up(q),Fk)anm,E),
so applying the third, first and fourth part of the proposition to the first, second and fourthterms in this sequence respectively we get a short exact sequence
0 // H1red(Up(q),Fk)m,E
// H1D(Up(q),Fk)m,E
// H0(g, Xq(Up(q),Fk)(−1)anm,E) // 0,
so identifying this with the short exact sequence of Proposition 2.3.2 gives the desired
isomorphism
H0(g, Xq(Up(q),Fk)
anm,E) ∼= Xq(U
p(q),Fk)m,E.
Finally the sixth isomorphism follows from the first four in the same way as the fifth
(using the second short exact sequence of Proposition 4.4.17.
Given a continuous G(Qp)-representation V we denote by V alg the space of locally
algebraic vectors in V . Let E(1) denote a one dimensional E-vector space on which
G(Qp) × Gal(L/L) acts by∏
v|pNFv/Qp ◦ det⊗ε, where ε is the unramified character
of Gal(L/L) sending Frobλ to NF/Q(λ). For any n ∈ Z let E(n) = E(1)⊗n. We can now
proceed as in Theorem 7.4.2 of [18] to deduce
Chapter 4. Completed cohomology of Shimura curves 79
Theorem 4.5.3. There are natural G(Qp)×Gq × T(Up(q))-equivariant isomorphisms
1.⊕
k,n∈ZH1D(Up(q),Fk)m,E ⊗E W∨
k,p ⊗E E(n) ∼= H1D(Up(q),Op)
algm,E.
2.⊕
k,n∈ZH1red(Up(q),Fk)m,E ⊗E W∨
k,p ⊗E E(n) ∼= H1red(Up(q),Op)
algm,E.
3.⊕
k,n∈ZH1red(Up(q), r∗r
∗Fk)m,E ⊗E W∨k,p ⊗E E(n) ∼= H1
red(Up(q), r∗r∗Op)
algm,E.
4.⊕
k,n∈ZXq(Up(q),Fk)m,E ⊗E W∨
k,p ⊗E E(n) ∼= Xq(Up(q),Op)
algm,E.
5.⊕
k,n∈Z Xq(Up(q),Fk)m,E ⊗E W∨
k,p ⊗E E(n) ∼= ˜Xq(Up(q),Op)
algm,E.
In all the above direct sums, the index k runs over d-tuples with all entries the same parity
and ≥ 2.
Proof. The proof is as for Theorem 7.4.2 of [18], since
{(ξ(k))∨ ⊗ (⊗di=1 det ◦ξi)n}k,n
is a complete set of isomorphism class representatives of irreducible algebraic representa-
tions of G which factor through Gc.
Applying the same arguments to G′, with m′ a non-Eisenstein maximal ideal of T′(V p)
we also obtain
Theorem 4.5.4. There are natural G′(Qp)×Gq × T′(V p)-equivariant isomorphisms
1.⊕
k,n∈ZH1D′(V
p,Fk)m′,E ⊗E W∨k,p ⊗E E(n) ∼= H1
D′(Vp,Op)
algm′,E.
2.⊕
k,n∈Z Yq(Vp,Fk)m′,E ⊗E W∨
k,p ⊗E E(n) ∼= Yq(Vp,Op)
algm′,E.
3.⊕
k,n∈Z Yq(Vp,Fk)m′,E ⊗E W∨
k,p ⊗E E(n) ∼= ˜Yq(V p,Op)algm′,E.
4.5.5 Cofreeness results
We now give analogues of Corollary 5.3.19 in [20].
Proposition 4.5.6. Choose Up small enough so that Up is pro-p and UpUp(q) is neat. Then
for each s > 0, H1D(Up(q),Op/$
s)m, H1red(Up(q),Op/$
s)m and H1red(Up(q), r∗r
∗Op)m
are injective as smooth representations of Up/F× ∩ UpUp over Op/$s.
4.5 Eigenvarieties and an overconvergent Jacquet–Langlands correspondence 80
In the statement of the proposition, we write Up/F× ∩ UpUp to denote the quotient of
Up by the closure in Up of the projection of F× ∩ UpUp to its factor at the place p.
Proof. We follow the proof of Proposition 5.3.15 in [20], with modifications due to the
presence of infinitely many global units O×F . Let L be any finitely generated smooth rep-
resentation of Up/F× ∩ UpUp = Up/F× ∩ UpUp(q) over Op/$s, with Pontriagin dual
L∨ := HomOp/$s(L,Op/$s). There is an induced local system L ∨ on each of the Shimura
curves M(U ′pUp(q)) as U ′p varies over the open normal subgroups of Up. As in Proposition
5.3.15 of [20] there is a natural isomorphism
H1(M(UpUp(q)),L ∨)m ∼= Hom
Up/F×∩UpUp(L,H1D(Up(q),Op/$
s)m).
Now starting from a short exact sequence 0 → L0 → L1 → L2 → 0 of finitely generated
smooth Up/F× ∩ UpUp-representations over Op/$s we obtain a short exact sequence of
sheaves on M(UpUp(q)):
0→ L ∨2 → L ∨
1 → L ∨0 → 0.
Taking the associated long exact cohomology sequence and localising at m gives anothershort exact sequence
0→ H1(M(UpUp(q)),L ∨
2 )m → H1(M(UpUp(q)),L ∨
1 )m → H1(M(UpUp(q)),L ∨
0 )m → 0,
or equivalently
0→ HomUp/F×∩UpUp(L2, H
1D(Up(q),Op/$
s)m)→ HomUp/F×∩UpUp(L1, H
1D(Up(q),Op/$
s)m)→
→ HomUp/F×∩UpUp(L0, H
1D(Up(q),Op/$
s)m)→ 0.
Therefore we conclude that H1D(Up(q),Op/$
s)m is injective as a smooth representation of
Up/F× ∩ UpUp over Op/$s. The same argument applies to the other cohomology spaces
(using Lemma 4.4.22 for the final case).
Definition 4.5.7. We say that a representation V of a topological group Γ overOp is cofree
if V ∼= C (Γ,Op)r for some integer r.
Corollary 4.5.8. If Up is small enough so that Up is pro-p and UpUp(q) is neat, then
H1D(Up(q),Op)m, H1
red(Up(q),Op)m and H1red(Up(q), r∗r
∗Op)m are all cofree representa-
tions of Up/F× ∩ UpUp.
Chapter 4. Completed cohomology of Shimura curves 81
Proof. The proof of Corollary 5.3.19 in [20] goes through.
Corollary 4.5.9. If Up is small enough so that Up is pro-p and UpUp(q) is neat, then
Xq(Up(q),Op)m is a cofree representation of Up/F× ∩ UpUp.
Proof. Taking the first exact sequence of Proposition 4.4.17 and applying the functor
Homcts(−,Op) gives a short exact sequence (we can ignore the Tate twist for the purposes
of this Corollary)
0→ Homcts(Xq(Up(q),Op)m,Op)→ Homcts(H
1D(Up(q),Op)m,Op)
→ Homcts(H1red(Up(q),Op)m,Op)→ 0.
Corollary 4.5.8 implies that the second and third non-zero terms in this sequence are free
Op[[Up/F× ∩ UpUp]]-modules, so Homcts(Xq(Up(q),Op)m,Op) is projective, hence free
since Op[[Up/F× ∩ UpUp]] is local. Applying the functor Homcts(−,Op) again gives the
desired result.
Similarly we obtain
Corollary 4.5.10. If Vp is small enough so that Vp is pro-p and VpVp is neat, then
H1D′(V
p,Op)m′ , Yq(Vp,Op)m′ and ˜Yq(V p,Op)m′ are cofree representations of Vp/F× ∩ VpV p.
4.5.11 Jacquet-Langlands maps between eigenvarieties
We can now apply the results of section 4.4.30 to deduce some cases of overconvergent
Jacquet-Langlands functoriality, or in other words maps between eigenvarieties interpolat-
ing the classical Jacquet-Langlands correspondence. Let T denote the rigid analytic variety
parametrising continuous characters χ : T → C×p .
Definition 4.5.12. Suppose V is a $-adically admissible G(Qp)-representation over Op,
with a commuting Op-linear action of a commutative Noetherian Op-algebra A. The es-
sentially admissible locally analytic T -representation JB(V anE ) gives rise (by duality) to
a coherent sheaf M on T , and the action of A gives rise to a coherent sheaf on T of
Op-algebras A ↪→ EndT (M ). Define the Eigenvariety
E (V,A)
to be the rigid analytic space given by taking the relative spectrum of A over T .
4.5 Eigenvarieties and an overconvergent Jacquet–Langlands correspondence 82
Lemma 4.5.13. Suppose V and A are as in Definition 4.5.12, further suppose that V is a
(non-zero) cofree representation of Up/X for some compact open subgroup Up of G(Qp),
X some closed subgroup of Z(Qp) ∩ Up. Then E (V,A) is equidimensional of dimension
equal to that of Up/X .
Proof. This is a combination of a mild generalisation of the proof of Proposition 4.2.36 in
[17] and Corollary 3.4.1 in Chapter 3 above.
In our applications we will have Up/X = Up/F× ∩ UpUp for some tame level Up. The
dimension of Up/X will therefore depend on the Leopoldt conjecture for F . Applying the
above lemma to the cofreeness results of the previous section, we have the following:
Corollary 4.5.14. The following eigenvarieties are all equidimensional:
• E (H1D(Up,Op)m,T(Up)m)
• E (Xq(Up(q),Op)m,T(Up(q))m)
• E (H1D′(V
p,Op)m′ ,T′(V p)m′)
• E (Yq(Vp,Op)m′ ,T′(V p)m′)
• E ( ˜Y q(Vp,Op)m′ ,T′(V p)m′)
Lemma 4.5.15. The embedding Xq(Up(q),Op)→ H1
D(Up(q),Op) induced by composing
the (monodromy pairing) embedding Xq(Up(q),Op) ↪→ ˜Xq(U
p(q),Op) with the embed-
ding ˜Xq(Up(q),Op) → H1
D(Up(q),Op) provided by Proposition 4.4.17 induces an iso-
morphism of eigenvarieties
E (Xq(Up(q),Op)m,T(Up(q))m) ∼= E (H1
D(Up(q),Op)q−newm ,T(Up(q))m).
Proof. This follows from Proposition 4.4.25.
We have a similar result for the group G′.
Lemma 4.5.16. The embedding Yq(Vp,Op)→ H1
D′(Vp,Op)induced by composing the (mon-
odromy pairing) embedding Yq(Vp,Op) ↪→ ˜Y q(V
p,Op)with the embedding ˜Y q(Vp,Op) ↪→
H1D′(V
p,Op)provided by Proposition 4.4.32 induces an isomorphism between eigenvari-
eties
E (Yq(Vp,Op)m′ ,T′(V p)m′) ∼= E (H1
D′(Vp,Op)m′ ,T′(V p)m′).
Chapter 4. Completed cohomology of Shimura curves 83
Proof. This follows from Proposition 4.4.33.
Lemma 4.5.17. The natural embedding Xq2(Up(q1q2),Op)q1−newm → H1
D(Up(q),Op)q1q2−newm
induces an isomorphism between eigenvarieties
E (Xq2(Up(q1q2),Op)q1−newm ,T(Up(q1q2))m) ∼= E (H1
D(Up(q1q2),Op)q1q2−newm ,T(Up(q1q2))m).
Proof. This follows from Proposition 4.4.35.
We now put ourselves in the situation of 4.4.30, so Up and V p are isomorphic at places
away from q1 and q2, and m and m′ give rise to the same irreducible mod p Galois repre-
sentation ρ.
Theorem 4.5.18. (Overconvergent Jacquet-Langlands correspondence) The embedding
Yq1(Vp,Op)ρ → Xq2(U
p(q1q2),Op)ρ
given by Theorem 4.4.34 induces an isomorphism between eigenvarieties
E (H1D′(V
p,Op)ρ,T′(V p)ρ) ∼= E (H1D(Up(q1q2),Op)
q1q2−newρ ,T(Up(q1q2))ρ).
Proof. It is clear that Theorem 4.4.34 induces an isomorphism E (Yq1(V p,Op)ρ,T′(V p)ρ) ∼=
E (Xq2(Up(q1q2),Op)q1−newρ ,T(Up(q1q2))ρ). Our theorem now follows by applying Lemmas
4.5.16 and 4.5.17.
Finally we note that level raising results in the same spirit as those in the first two
chapters follow from the equidimensionality of the eigenvarieties described in this section.
In particular we have
Corollary 4.5.19. The following eigenvarieties are equidimensional:
• E (H1D(Up(q),Op)
q−newρ ,T(Up(q))ρ)
• E (H1D(Up(q1q2),Op)
q1q2−newρ ,T(Up(q1q2))ρ)
Proof. The first claim follows from the second part of Corollary 4.5.14 and Lemma 4.5.15.
The second claim follows from the fourth part of Corollary 4.5.14 and Theorem 4.5.18.
84
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